
\count100= 1
\count101= 15
   %%replace 3 by the number of pages of your paper.
\input at03.tex
%\input lls00.tex  %% Remove the % sign if you want advanced macros
%\draft  \def\updated{August 20, 2003}
   %% Remove the % sign to get a draft version showing assigned labels
   %% This only works if you input lls00.tex


\title{Pseudometrics, Distances and }
\titwo{Multivariate Polynomial Inequalities}
  %% If need two lines for title, replace xxx.
\author{Len Bos, Norm Levenberg and Shayne Waldron}
 
%\autwo{xxx}
  %% If you need two lines for authors, replace xxx.

\def\shorttitle{Pseudometrics, Distances and Multivariate Polynomial Inequalities}
  %% Replace xxx by a short title for the running head.
\def\shortauthor{L.~Bos, N.~Levenberg and S.~Waldron}
   %% Replace xxx by author names (initials only) for running head

%   ********************** MACROS  ******************************

%%  Insert your own macros here


%************************* ABSTRACT **********************************

\abstract {We discuss three natural pseudodistances and pseudometrics on a bounded 
domain in $\RR^N$ based on polynomial inequalities.}
\vskip6pt
{\bf Key words and phrases}: pseudodistance, pseudometric, polynomial inequalities
\vskip6pt
{\bf AMS-MOS classification numbers}: Primary 41A17; Secondary 41A63, 26D10  

%******************************* BODY *********************************

\sect{0. Introduction.} In [3], for a compact set $K\subset\RR^n$, we defined a 
Carath\'eodory type distance due to Dubiner [6] and
a Finsler type distance based on Baran's generalization
of the van der Corput - Schaake polynomial inequality [1], [2]. 
These distances are intimately connected to the distribution of 
optimal points for multivariate polynomial interpolation, as well
as to the distribution of nodes for ``good'' quadrature rules (cf., the Introduction 
and the references of [3]).

Let $K=\overline \Omega\subset \RR^N$ where $\Omega$ is a domain. We expand upon the definitions given in [3] 
in proving some general relationships among {\it three} natural pseudodistances 
as well as three natural pseudometrics on $\Omega$.  

The classical Markov inequality, or more precisely, the van der Corput - Schaake inequality, says that  
for $p:\RR\to \RR$ a real
polynomial such that $||p||_I=\sup_{x\in [-1,1]}|p(x)|\le 1$,  
$$\left|{p'(x)\over\sqrt{1-p^2(x)}}\right|\le\hbox{deg}(p){1\over\sqrt{1-x^2}}, \ x\in (-1,1).$$
This is equivalent to
$${1\over \hbox{deg}(p)}
\left|{d\over dx}\cos^{-1}(p(x))\right|\le\left|{d\over dx}\cos^{-1}(x)\right|$$
which motivates the definition of the {\it Dubiner} pseudodistance (Definition 1.4). 

Analogously, estimates on ${1\over {\rm deg}p}{|D_yp(x)|\over \sqrt {1-p(x)^2}}$ for polynomials $p:\RR^N\to \RR$ normalized 
with $||p||_K=\sup_{x \in K}|p(x)|\leq 1$, where $D_yp(x)$ denotes the directional derivative of $p$ at $x$ in the direction $y$, give rise to the definition 
of the {\it Markov} pseudometric (Definition 1.5).

Next, we recall for a compact set $K\subset \CC^N$, the function $$V_K(z):=\sup\{\log {|p(z)|^{1/{\rm
deg}(p)}}\,:\,p: \CC^N\to \CC,\,
\hbox{deg}(p)\ge 1,\,||p||_K\le1\}$$
is known as the Siciak-Zaharjuta extremal function. If $V_K(z)$ is finite, which it is for all $z\in \CC^N$ when 
$K=\overline \Omega\subset \RR^N$ where
$\Omega$ is a domain, then for any polynomial $p$ and any point
$z$, from the definition of $V_K$ we have the Bernstein-Walsh inequality
$$|p(z)|\le e^{{\rm deg}(p)V_K(z)}||p||_K. $$
The function $V_K$ will be utilized, in particular, in defining and analyzing the {\it Baran} pseudometric and pseudodistance (Definition 1.6).

The organization of the paper is as follows: in section 1, we define the notions of pseudometric and pseudodistance on domains in $\RR^N$. We follow
closely the presentation in Jarnicki-Pflug [7], but we also recommend Dineen's monograph [5]. Then we define the Dubiner, Markov and Baran
pseudodistances and pseudometrics for a bounded domain $\Omega \subset \RR^N$ and recall the results of the relevant calculations from [3]. In section 2, we give relationships among these 
pseudodistances and pseudometrics for general $\Omega$ and we prove certain properties (monotonicity, invariance, etc.). Finally, in the last section, we show that
all three pseudometrics coincide when $K=\overline \Omega$ is a symmetric convex body in $\RR^N$ (Proposition 3.6). The corresponding pseudodistances are shown to
coincide for symmetric convex bodies in 
$\RR^2$ that satisfy a technical condition; we conjecture that this additional condition is not needed, and that indeed
the result is true in $\RR^N.$ This is {\it
not} the case, in general, for non-symmetric convex bodies as 
was shown in [3] via the example of the simplex in $\RR^2$. 



\vskip10pt

\sect{1. Definition of the pseudodistances and pseudometrics.} 
We begin our discussion with the definitions of pseudodistances 
and pseudometrics; we refer the reader to section 4.3 of [7] for details and proofs of Propositions 1.1-1.6. A word of warning: in [7], the field of scalars is
$\CC$. Let
$K=\overline
\Omega\subset
\RR^N$ where
$\Omega$ is a domain. 

\proclaim Definition 1.1. \ We call 
$F:\Omega \times \RR^N\to \RR^+$ a {\it pseudometric} if
\item {a.} $F(x;\lambda)$ is uppersemicontinuous (usc) as a function of $(x,\lambda)\in \Omega \times \RR^N$;
\item {b.} $F(x;\lambda)$ is positive definite in $\lambda$: $F(x;\lambda)\geq 0$ and $F(x;\lambda)=0$ if and only if $\lambda=0$;
\item {c.} $F(x;\lambda)$ is positively homogeneous in $\lambda$: $F(x;t\lambda)=|t|F(x;\lambda)$ for $t\in \RR$.

\noindent {\bf Remark 1.1}. It follows from a. and c. that 
{\sl \item {d.} $F(x;\lambda)$ is locally Lipschitz in $\lambda$: $F(x;\lambda) \leq c|\lambda|$ where $c=c(x)$ depends on $x$ and is locally bounded above.}

More precisely, we should call $F$ an {\it usc} pseudometric; but we omit the adjective usc. All of our pseudometrics will, in addition, satisfy a bi-Lipschitz
condition:
{\sl \item {d$'$.} $c_1|\lambda| \leq F(x;\lambda) \leq c_2|\lambda|$ where $c_i=c_i(x), \ i=1,2$ depend on $x$ with $c_1$ locally bounded below and $c_2$ locally bounded above.}
\vskip6pt

\proclaim Definition 1.2. \ We call $d:\Omega \times \Omega\to \RR^+$ a {\it pseudodistance} if 
\item {A.} $d(a,b)=d(b,a)\geq 0$;
\item {B.} $d(a,b)\leq d(a,c)+d(c,b)$;
\item {C.} $d$ is locally dominated by the Euclidean distance: for all $c\in \Omega$ there 
exists $M>0, \ r>0$ with $d(a,b)\leq M|a-b|$ if $a,b\in \Omega$ with $\max  \{|a-c|,|b-c|\}<r$.
\vskip4pt

All of our pseudodistances will {\it locally dominate} the Euclidean distance; hence:
{\sl \item {D.} for all $c\in \Omega$ there 
exist $m,M>0, \ r>0$ with $m|a-b|\leq d(a,b)\leq M|a-b|$ if $a,b\in \Omega$ with $\max  \{|a-c|,|b-c|\}<r$.}
\vskip4pt

If $d(a,b)>0$ for $a\neq b$, we call $d$ a {\it distance}; from D., all of our pseudodistances will be distances.
\vskip6pt 

We summarize four operations with $d,F$:

\item {1.} {\sl The operator $d\to d^i$}:

Given a pseudodistance $d$, let $\alpha:[0,1]\to \Omega$ denote a continuous curve. Define
$$L_d(\alpha):=\sup\{\sum_{j=1}^nd(\alpha(t_{j-1}),\alpha(t_j)):0=t_0<\cdots < t_n=1\},$$
the $d-$length of $\alpha$. Define $d^i:\Omega \times \Omega\to \RR^+$ via
$$d^i(a,b):=\inf \{L_d(\alpha):\alpha \ \hbox{continuous curve in} \ \Omega \ \hbox{joining} \ a,b\}.$$
We call $d^i$ the {\it inner pseudodistance} associated to $d$.
\proclaim Proposition 1.1. \ $d^i$ is a pseudodistance; $d\leq d^i$; and $L_{d^i}=L_d$. \par

\item {2.} {\sl The operator $F\to \int F$}:

Given a pseudometric $F$ and $\alpha:[0,1]\to \Omega$ a piecewise $C^1$ curve, define
$$L_F(\alpha):=\int_0^1F(\alpha(t);\alpha'(t))dt,$$
the $F-$length of $\alpha$. Define, for $a,b\in \Omega$,
$$(\int F)(a,b):= \inf \{L_F(\alpha):\alpha \ \hbox{piecewise $C^1$ curve in} \ \Omega \ \hbox{joining} \ a,b\}.$$
\proclaim Proposition 1.2. \ $\int F$ is a pseudodistance; and $L_{\int F}\leq L_F$ for each piecewise $C^1$ curve; hence $(\int F)^i=\int F$. \par

\item {3.} {\sl The operator $d \to Dd$}:

Given a pseudodistance $d$, define, for $x\in \Omega$ and $y\in \RR^N$,
$$Dd(x;y):=\limsup_{t\to 0^+, \ z\to x}{d(z,z+ty)\over t}.$$
\proclaim Proposition 1.3. \ $Dd$ is a pseudometric;
\item {(i)} $Dd(x;y):=\limsup_{x_1,x_2\to x, \ {x_1-x_2\over |x_1-x_2|}\to y} {d(x_1,x_2)\over |x_1-x_2|}, \ |y|=1$;
\item {(ii)} $d\leq \int (Dd)$; 
\item {(iii)} for any pseudometric $F$, $D(\int F)\leq F$. \par

\item {4.} {\sl The operator $F \to \widehat F$}:

Given $f:\RR^N\to \RR^+$ satisfying $f(tx)=|t|f(x)$ for $t\in \RR$ and $x\in \RR^N$, and $f(x)\leq M|x|$, define 
$$\Gamma (f):=\{y\in \RR^N: |y\cdot z|=|\sum_{j=1}^N y_jz_j|\leq f(z), \ \hbox{for all} \ z\in \RR^N\}$$
$$=\{y\in \RR^N: y\cdot z\leq f(z), \ \hbox{for all} \ z\in \RR^N\}$$
$$=\{y\in \RR^N: y\cdot z\leq 1, \ \hbox{for all} \ z\in \RR^N \ \hbox{with} \ |f(z)|=1\};$$
the first equality occurs since $f(-x)=f(x)$, the second from $f(tx)=|t|f(x)$. The (filled-in) indicatrix of such an absolutely homogeneous $f$ is the set
$$E=\{x\in \RR^N: f(x)\leq 1\};$$
and the polar of a set $E\subset \RR^N$ is the set
$$E^*:=\{y\in \RR^N: y\cdot z\leq 1, \ \hbox{for all} \ z\in E\};$$
thus $\Gamma (f)$ is the polar of the ``filled-in'' indicatrix of $f$. Next, define 
$$\widehat f(x):= \sup \{x\cdot y: y\in \Gamma(f)\};$$
this is the support function of $\Gamma(f)$. Note that $f\leq g$ implies $\hat f\leq \hat g$. Recalling that an absolutely homogeneous $f$ defines a {\it seminorm}
if $f(x+y)\leq f(x) +f(y)$, we have  
\item {a.} $\widehat f \leq f$;
\item {b.} $\widehat f$ is always a seminorm and $f$ is a seminorm iff $\widehat f=f$;
\item {c.} $\Gamma (\widehat f)=\Gamma(f)$;
\item {d.} $\{\widehat f \leq 1\}$ is the closed convex hull of $\{f\leq 1\}$. 

\noindent (cf., Remark 4.3.4 in [7]). Now given a pseudometric $F$, define $\widehat F(x;y):= F(x;\widehat \cdot)$ (``hat'' operation in second variable).

\proclaim Proposition 1.4. \ $\widehat F$ is a pseudometric and $\int \widehat F =\int F$. Moreover, 
$D(\int F)\leq \widehat F$; for $F$ satisfying $d'$ (of Remark 1.1), we have equality if $F$ is continuous in $(x;y)$.

\proclaim Proposition 1.5. \  We have the following relations between the operations $d^i, \ \int, \ Dd, \ \widehat F$:
\item \item {(i)} $D(\int F)\leq \widehat F$;
\item \item {(ii)} $\int (Dd)\geq d^i$;
\item \item {(iii)} $\int (\widehat F) =\int F$;
\item \item {(iv)} $\widehat {Dd}=Dd$. \par

For use in section 3, we define the notion of a {\it $C^1$ pseudodistance}. Below, $B(x,r)$ denotes the Euclidean ball of radius $r$ centered at $x$.
\vskip6pt

\proclaim Definition 1.3 ($C^1$ pseudodistance). A pseudodistance $d$ on $\Omega$ is a {\it $C^1$ pseudodistance} if for all $E\subset \subset \Omega$, and all
$\epsilon >0$, there  exists $\eta >0$ such that
$$|d(x_1,x_2)-(Dd)(x;x_1-x_2)|\leq \epsilon |x_1-x_2|$$
for $x\in E$ and $x_1,x_2\in B(x,\eta)$. \par

\proclaim Proposition 1.6. \ Let $d$ be a $C^1$ pseudodistance. Then $d^i=\int (Dd)$ and $d^i$ is a $C^1$ pseudodistance. \par

\vskip6pt

We now define our natural pseudodistances and pseudometrics on a bounded domain $\Omega \subset \RR^N$. The applications we have in mind and some of 
the fundamental notions we utilize deal with compact sets; thus we often consider one or more of the six items below as associated to $K=\overline \Omega$.  


\proclaim Definition 1.4 (Dubiner pseudodistance and pseudometric). 
$$d_D^K(a,b)=d_D(a,b):=\sup_{||p||_K\leq 1, \ {\rm deg}p\geq 1} {1\over {\rm deg}p}|\cos^{-1}(p(a))-\cos^{-1}(p(b))|$$
is the {\it Dubiner pseudodistance on $K$}. Note that this is well-defined on $K\times K$ for any compact set $K$. For $x\in \Omega$ and $y\in \RR^N$,
$$\delta_D^K(x;y)=\delta_D(x;y):=Dd_D(x;y):=\limsup_{t\to 0^+, \ z\to x}{d_D(z,z+ty)\over t}$$
is the {\it Dubiner pseudometric for $K$}.  
\vskip6pt

\proclaim Definition 1.5 (Markov pseudodistance and pseudometric). 
$$\delta_M^K(x;y)=\delta_M(x;y):=\sup_{||p||_K\leq 1, \ {\rm deg}p\geq 1}{1\over {\rm deg}p}{|D_yp(x)|\over \sqrt {1-p(x)^2}},$$
(for $x\in \Omega$ and $y\in \RR^N$) defined for compacta $K$ for which it is usc, is the {\it Markov pseudometric for $K$} and
$$d_M^K=d_M =\int \delta_M$$
is the {\it Markov pseudodistance for $K$}. From the results of [2], $\delta_M^K$ is continuous at $x\in\Omega=K^o$
if $K$ is a centrally symmetric convex body (see Cor. 3.5).
\vskip6pt

\proclaim Definition 1.6 (Baran pseudodistance and pseudometric). 
$$\delta_B^K(x;y)=\delta_B(x;y):=\limsup_{t\to 0^+}{V_K(x+ity)\over t},$$
(for $x\in \Omega$ and $y\in \RR^N$) defined for compacta $K$ for which it is usc, is the {\it Baran pseudometric for $K$} and
$$d_B^K=d_B =\int \delta_B$$
is the {\it Baran pseudodistance for $K$}. From the results of [4], $\delta_B^K$ is continuous for $x\in K^o$ if $K$ 
is an arbitrary convex body. Moreover, in this case, the limit in the definition of $\delta_B^K$ exists. 
\vskip6pt

\noindent {\bf Remark 1.2}. When the set $K$ is understood, we delete the
superscript $K$ for our pseudodistances and pseudometrics. 
\vskip6pt

\noindent {\bf Remark 1.3}. For the unit cube $C$ in $\RR^N$, one can explicitly compute 
$$\delta_M^C(x;y)=\delta_B^C(x;y)=\max_{j=1,...,N}{|y_j|\over \sqrt {1-x_j^2}}$$
(see [3]). Since $K_1\subset K_2$ clearly implies $\delta_M^{K_1}(x;y)\geq \delta_M^{K_2}(x;y)$ and 
$\delta_B^{K_1}(x;y)\geq \delta_B^{K_2}(x;y)$ for $x$ in the interior of $K_1$,  
for any $K=\overline \Omega$ in $\RR^N$ we see by taking a cube inside $K$ and another containing $K$ that 
$\delta_M^K$ and $\delta_B^K$ are pseudometrics satisfying the bi-Lipschitz property $d'$. Proposition 1.2 shows that 
$d_M^K$ and $d_B^K$ are pseudodistances; i.e., they satisfy A.-C. of Definition 1.2. The fact that $\delta_D^K$ is a 
pseudometric satisfying the bi-Lipschitz property $d'$ will follow from Proposition 2.1 (equation (2.2)). Finally, the 
verification of property C. of Definition 1.2 for $d_D^K$  -- A. and B. are trivial -- will follow from Proposition 2.1 (equation (2.1)). 
To verify the other half of property D. for $d_D^K$, take $r>0$ so that 
the Euclidean ball $B(c,r)\subset \Omega$ and let $p$ be the polynomial of degree one which is (normalized) 
linear projection to the line joining $a$ and $b$. Proposition 2.1  (equation (2.1)) will imply the same property for $d_M^K$ and $d_B^K$. 
\vskip6pt

\noindent {\bf Remark 1.4}. We see from Proposition 1.2 that each of the Markov and Baran pseudodistances are inner; i.e., 
$d_M^i=d_M$ and $d_B^i=d_B$.
\vskip6pt

As concrete examples, we summarize the following calculations in [3]:
\item {(i)} For $K=\overline \Omega =\{x=(x_1,...,x_N)\in\RR^N: |x|^2=\sum_{j=1}^N x_j^2 \leq 1\}$ 
the closed unit ball,  $d_D(a,b)=d_B(a,b)=\cos^{-1}(\tilde a \cdot \tilde b)$ 
where $\tilde a=(a,\sqrt {1-|a|^2}), \  \tilde b=(b,\sqrt {1-|b|^2})$ are the liftings of $a,b$ to 
the surrounding unit sphere $S^N\subset \RR^{N+1}$. From Proposition 2.1 in the next section, we conclude that $d_D(a,b)=d_M(a,b)=d_B(a,b)$.
\item {(ii)} For $K=\overline \Omega =I^N=\{x=(x_1,...,x_N)\in\RR^N: \max_{j=1,...,N} |x_j| \leq 1\}$ 
the closed unit cube,  $d_D(a,b)=d_B(a,b)=\max_{j=1,...,N}d^I_D(a_j,b_j)=\max_{j=1,...,N}|\cos^{-1}b_j - \cos^{-1}a_j|$. 
>From Proposition 2.1 in the next section, we conclude that $d_D(a,b)=d_M(a,b)=d_B(a,b)$.
\item {(iii)} For $K=\overline \Omega =\{x=(x_1,...,x_N)\in\RR^N: x_j \geq 0, \ \sum_{j=1}^N x_j\leq 1\}$ 
the standard simplex, $d_B(a,b)=2[\cos^{-1}(\tilde a \cdot \tilde b)]$. Here, $d_D(a,b)\not \equiv d_B(a,b)$. 


\vskip10pt

\sect{2. Pseudodistances and pseudometrics: general $K$.} 
In this section, we let $K=\overline \Omega\subset \RR^N$ where $\Omega$ is a bounded 
domain such that $\delta_M$ and $\delta_B$ are usc, and we derive the following inequalities relating the Dubiner, Markov and Baran pseudodistances and pseudometrics. 

\proclaim Proposition 2.1. For $K=\overline \Omega\subset \RR^N$ we have
$$d_D\leq d_D^i \leq d_M\leq d_B \eqno(2.1)$$
and
$$\delta_D=\delta_M\leq \delta_B. \eqno(2.2)$$
\par

\noindent {\bf Proof}. First note that for any polynomial $p$ with 
$||p||_K\leq 1$, and any two points $a,b\in \Omega$, if $\alpha:[0,1]\to \Omega$ is a $C^1$ curve with 
$\alpha(0)=a$ and $\alpha (1)=b$, then
$$|\cos^{-1}(p(b))-\cos^{-1}(p(a))|=|\int_0^1{d\over dt}[\cos^{-1}(p(\alpha(t))]dt|$$
$$\leq 
\int_0^1{\bigl |(D_{\alpha'(t)}p(\alpha(t))\bigr |\over \sqrt {1-(p(\alpha(t)))^2}}dt.$$
Taking the supremum over all such polynomials and the infimum over all 
such $C^1$ curves shows that $d_D\leq d_M$. Moreover, we have $$\int_0^1{\bigl |(D_{\alpha'(t)}p(\alpha(t))\bigr |\over \sqrt
{1-(p(\alpha(t)))^2}}dt
\leq {\rm deg}p\int_0^1\delta_B(\alpha(t);\alpha'(t))dt.$$ This last inequality is {\it Baran's inequality} (Theorem 1.14 of [1]) and actually holds with
$\delta_B(x;y)$ replaced by
$$\tilde \delta_B(x;y):=\liminf_{t\to 0^+}{V_K(x+ity)\over t}.$$
In particular, we get $\delta_M\leq \delta_B$ and hence, from the definitions of the Markov and Baran pseudodistances, that $d_M\leq d_B$. It follows that 
$$d_D\leq d_M\leq d_B. \eqno(2.3)$$
We also conclude that 
$$Dd_D\leq Dd_M\leq Dd_B. \eqno(2.4)$$

Next we show that
$$\delta_M\leq Dd_D. \eqno(2.5)$$
For, by definition of $d_D$, for any polynomial $p$ with 
$||p||_K\leq 1$,
$${1\over {\rm deg}p}{|\cos^{-1}(p(x+ty))-\cos^{-1}(p(x))|\over t}\leq {d_D(x+ty,x)\over t}.$$
Thus
$${1\over {\rm deg}p}{|D_yp(x)|\over \sqrt {1-(p(x))^2}}\leq \limsup_{t\to 0^+}{d_D(x+ty,x)\over t}\leq (Dd_D)(x;y).$$
Hence $\delta_M\leq Dd_D$. Combining (2.4) and (2.5) we have
$$\delta_M\leq Dd_D\leq Dd_M\leq Dd_B. \eqno(2.6)$$
 

Now from Proposition 1.5, $\widehat {Dd}=Dd$ (property (iv)), and $D(\int F)\leq \widehat F$ (property (i)); thus, taking ``hats'' of (2.6), 
$$\widehat {\delta_M}\leq \widehat {Dd_D}=Dd_D\leq \widehat {Dd_M}=Dd_M\leq \widehat {\delta_M}.$$
Thus equality holds throughout and, in particular, 
$$Dd_M= \widehat {\delta_M}.$$
But $\widehat {\delta_M}\leq \delta_M \leq Dd_D=\widehat {\delta_M}$ so that
$$\delta_D=Dd_D= Dd_M= \widehat {\delta_M}=\delta_M. \eqno(2.7)$$
Together with (2.6) and the conclusion from Baran's inequality that $\delta_M\leq \delta_B$, this completes the proof of (2.2). Finally, 
integrating (2.7) to get a relation among the pseudodistances, we have
$$d_D^i\leq \int Dd_D=\int Dd_M=\int \widehat {\delta_M}=\int \delta_M =d_M$$
using (ii) from Proposition 1.5. Together with (2.3), this completes the proof of (2.1).
 \hfill $\clubsuit$ \par
\vskip6pt

Based on Remark 1.3 and property D. of Definition 1.2, we delete the ``pseudo'' in referring henceforth to the Dubiner, Baran and Markov {\it distances}. We make a
few useful observations about the Dubiner distance and pseudometric.
\proclaim Lemma 2.2. \ For $a,b\in K$ and a positive integer $k$, we have 
$$d_D(a,b)=d_D^{(k)}(a,b):=\sup_{||p||_K\leq 1, \ {\rm deg}p\leq k} 
{1\over {\rm deg}p}|\cos^{-1}(p(b))-\cos^{-1}(p(a))|$$ for $k\geq {\pi \over
d_D^{(1)}(a,b)}$. \par

\noindent {\bf Proof}. If deg$p>k$, then ${1\over {\rm deg}p}|\cos^{-1}(p(a))-\cos^{-1}(p(b))|
\leq {\pi \over {\rm deg} p}<{\pi\over k}\leq d_D^{(1)}(a,b)$. \hfill
$\clubsuit$ \par
\vskip6pt

\proclaim Lemma 2.3. \ We have
$$\delta_D(x;y)=\lim_{t\to 0+}{d_D(x,x+ty)\over t},$$
i.e., the limit in the definition of the Dubiner pseudometric exists. \par

\noindent {\bf Proof}. Recall that 
$$\delta_D^K(x;y)=\delta_D(x;y):=Dd_D(x;y):=\limsup_{t\to 0^+, \ z\to x}{d_D(z,z+ty)\over t}.$$
By definition of $d_D$, for any polynomial $p$ with 
$||p||_K\leq 1$,
$${1\over {\rm deg}p}{|\cos^{-1}(p(x+ty))-\cos^{-1}(p(x))|\over t}\leq {d_D(x+ty,x)\over t}.$$
Thus
$${1\over {\rm deg}p}{|D_yp(x)|\over \sqrt {1-(p(x))^2}}=\liminf_{t\to
0^+} {1\over {\rm deg}p}{|\cos^{-1}(p(x+ty))-\cos^{-1}(p(x))|\over t}$$
$$\leq \liminf_{t\to
0^+}{d_D(x+ty,x)\over t}\leq \limsup_{t\to
0^+}{d_D(x+ty,x)\over t}\leq \delta_D(x;y).$$
By (2.2), $\delta_D(x;y)=\delta_M(x;y)$; moreover
the above inequality for any polynomial $p$ with 
$||p||_K\leq 1$ implies that 
$$\delta_M(x;y)\leq \liminf_{t\to
0^+}{d_D(x+ty,x)\over t};$$ combining these inequalities,
$$\delta_M(x;y)\leq \liminf_{t\to
0^+}{d_D(x+ty,x)\over t}\leq\limsup_{t\to
0^+}{d_D(x+ty,x)\over t}\leq \delta_M(x;y)$$
so that the limit exists. \hfill
$\clubsuit$ \par
\vskip6pt

Next we discuss invariance properties. We begin with the Dubiner distance.

\proclaim Lemma 2.4. \ For a polynomial map $P=(p_1,...,p_N):\RR^N\to \RR^N$ with deg$P:=\max ($deg$p_1,...,$deg$p_N)$ and $a,b\in K$,
$$d^K_D(a,b)\geq {1\over {\rm deg} P}d^{P(K)}_D(P(a),P(b)).$$
For an invertible linear map $T:\RR^N\to \RR^N$ and $a,b\in K$,
$$d^K_D(a,b)=d^{T(K)}_D(T(a),T(b)).$$ \par

\noindent {\bf Proof}. The inequality 
follows from the definition of $d^K_D$ and $d^{P(K)}_D$. In particular, this inequality holds for 
an invertible linear map $T:\RR^N\to \RR^N$. 
The reverse inequality in this case follows by applying the above inequality with $K,P(K)$ and the map $P$ replaced by the sets $T(K), T^{-1}(T(K))=K$ and 
the map $T^{-1}$.
\hfill
$\clubsuit$ \par
\vskip6pt

The Markov pseudometric is invariant under invertible linear maps.

\proclaim Lemma 2.5. \ For an invertible linear map $T:\RR^N\to \RR^N$, $\delta_M^K(x;y)=\delta_M^{T(K)}(T(x);T(y))$. \par

\noindent {\bf Proof}. First of all, clearly $\delta_M^K$ is usc if and only if $\delta_M^{T(K)}$ is usc (the same is true for $\delta_B$; this will be used in Corollary 2.7). From the definition,
$$\delta_M^K(x;y)=\sup_{||Q||_K\leq 1, \ {\rm deg}Q\geq 1}{1\over {\rm deg}Q}{|D_{y}Q(x)|\over \sqrt {1-Q(x)^2}}.$$
Now if $Q(x)=(p\circ T)(x)$, and we call $x'=T(x)$, then
$$D_yQ(x)=\nabla_x Q(x)\cdot y= T^t(\nabla_{x'}p(x'))\cdot y=\nabla_{x'}p(x')\cdot T(y)=D_{T(y)}p(T(x)).$$
Note that if $||p||_{T(K)}\leq 1$, then $||Q||_K\leq 1$. We obtain
$$\delta_M^{T(K)}(T(x);T(y))=\sup_{||p||_{T(K)}\leq 1, \ {\rm deg}p\geq 1}{1\over {\rm deg}p}{|D_{T(y)}p(T(x))|\over \sqrt {1-[p(T(x))]^2}}$$
$$\leq \sup_{||Q||_K\leq 1, \ {\rm deg}Q\geq 1}{1\over {\rm deg}Q}{|D_{y}Q(x)|\over \sqrt {1-Q(x)^2}}=\delta_M^K(x;y).$$
Applying the above argument with $T^{-1}$, we obtain
$$\delta_M^K(x;y)=\delta_M^{T^{-1}(T(K))}((T^{-1}\circ T)(x);(T^{-1}\circ T)y)\leq \delta_M^{T(K)}(T(x);T(y))$$
and equality holds. \hfill
$\clubsuit$ \par
\vskip6pt

Finally we turn to the Baran distance and pseudometric. 
We recall a result of Klimek [8, Thm.~5.3.1]: 
if $P=(p_1,...,p_N):\CC^N\to \CC^N$ is a proper polynomial mapping of degree $d$,
then $V_K(P(z))=d V_{P^{-1}(K)}(z)$.

\proclaim Lemma 2.6. \ Suppose 
$P=(p_1,...,p_N):\RR^N\to \RR^N$ is a polynomial mapping satisfying the Klimek condition: $d:=$deg$p_1=...=$deg$p_N$ and $\hat
P^{-1}(0)=\{0\}$ where
$\hat P$ is the homogeneous part of $P$ of degree $d$. Let $A,C \subset \RR^N$ 
with $C=P(A)$ and suppose that if $x\in A$ with det$JP(x)=0$, then $P(x)\in \partial C$. Then
$$\delta^A_B(x;y)={1\over d}\delta^C_B(P(x);JP(x)\cdot y)$$
and hence
$$d^A_B(a,b)={1\over d}d^C_B(P(a),P(b)).$$ \par

\noindent {\bf Proof}. 
%Note that if $\delta^A_B$ is usc, then so is $\delta^C_B$. 
Using Klimek's result, we have
$$\delta^A_B(x;y)=\limsup_{t\to 0^+}{V_A(x+ity)\over t}={1\over d}\limsup_{t\to 0^+}{V_C(P(x+ity))\over t}$$
$$={1\over d}\limsup_{t\to 0^+}{V_C(P(x)+JP(x)\cdot ity+0(t^2))\over t}={1\over d}\delta^C_B(P(x);JP(x)\cdot y).$$
Here, the last equality follows from the considerations of Remark 1.3.

Hence, letting $\gamma$ vary over curves in the interior $A^o$ of $A$ joining two points $a$ and $b$, and letting $\tilde \gamma$ vary 
over compositions $P\circ \gamma$,
$$d^A_B(a,b)=\inf_{\gamma}\int_0^1\delta^A_B(\gamma(t);\gamma'(t))dt$$
$$={1\over d}\inf_{\gamma}\int_0^1\delta^C_B(P(\gamma(t));JP(\gamma(t))\cdot \gamma'(t))dt$$
$$={1\over d}\inf_{\tilde \gamma}\int_0^1\delta^C_B(\tilde \gamma(t);\tilde \gamma'(t))dt$$
$$={1\over d}\inf_{\Gamma}\int_0^1\delta^C_B(\Gamma(t); \Gamma'(t))dt={1\over d}d^C_B(P(a),P(b)).$$
Here $\Gamma$ varies over all curves joining $P(a), P(b)$ and the first equality in the last line follows from our hypothesis that det$JP(x)\not =0$ if $P(x)\in
C^o$. \hfill
$\clubsuit$ \par
\vskip6pt

\proclaim Corollary 2.7. \ For an invertible linear map $T:\RR^N\to \RR^N$,
$$\delta^K(x;y)=\delta^{T(K)}(T(x);T(y)) \eqno(2.8) $$
for each of the pseudometrics $\delta = \delta_D,\delta_M$, or $\delta_B$; and
$$d^K(a,b)=d^{T(K)}(T(a),T(b)) \eqno(2.9)$$
for each of the distances $d = d_D,d_M$, or $d_B$. \par
\vskip6pt


\noindent {\bf Remark 2.1} For $K_1\subset K_2$,  $d^{K_2}\leq d^{K_1}$ on $K_1^o\times K_1^o$ for each of the distances $d_D, \ d_M, \ d_B$.

\vskip10pt

\sect{3. $K$ convex and centrally symmetric.} At the end of section 1 we noted that the three distances coincide on balls and cubes. In
this section, we study the connection between our three pseudometrics and distances for $K\subset \RR^N$ a 
{\it centrally symmetric convex
body}; i.e., $K$ is compact and convex with 
$\Omega =K^o\not = \emptyset$ and
$K=-K$.  Let $|||x|||_K:=\inf \{\lambda >0:x\in \lambda K\}$. Then $K$ is the closed unit ball in this norm:
$$K=\{x\in \RR^N: |||x|||_K\leq 1\}.$$

Motivated by some results due to Milev and R\'ev\'esz [10] in their investigation of the ``inscribed ellipse'' 
method of Sarantopoulos [12] (see also Kro\'o and R\'ev\'esz [9]) for investigating Markov inequalities
in convex bodies, we obtain a geometric interpretation of the Markov pseudometric in Lemma 3.2. 
This will be used to verify equality of the three pseudometrics in Proposition 3.6. To begin, given
$x\in K$ and
$y\in
\RR^N$, let
$$E_b(x,y):=\{r(t)=x\cos t+yb\sin t: 0\leq t \leq 2\pi\}.\eqno (3.1)$$
This is a centrally symmetric ellipse containing the points $\pm x, \pm yb$. The point of the ``inscribed ellipse'' method is to scale $b$ to fit inside $K$. 

\proclaim Lemma 3.1. \ For $b\leq {\sqrt {1-|||x|||_K^2}\over |||y|||_K}, \ E_b(x,y)\subset K$. \par

\noindent {\bf Proof}. We have
$$|||r(t)|||_K\leq |||x|||_K|\cos t| +|||y|||_K b|\sin t|$$
$$\leq |||x|||_K|\cos t| +\sqrt {1-|||x|||_K^2} |\sin t|\leq 1\cdot \sqrt {|||x|||_K^2+1-|||x|||_K^2}=1.$$ \hfill
$\clubsuit$ \par
\vskip6pt

Now let 
$$b^*(x,y):=\sup \{b:E_b(x,y)\subset K\}. \eqno(3.2)$$ 
By definition and Lemma 3.1,
$$b^*(x,y) \geq {\sqrt {1-|||x|||_K^2}\over |||y|||_K}.$$

\proclaim Lemma 3.2. \  Let $x\in K^o, \ y\in \RR^N$. For $p$ a polynomial with $||p||_K\leq 1$ and $|p(x)|\not = 1$, 
$${1\over {\rm deg}p}{|D_yp(x)|\over \sqrt {1-(p(x))^2}}\leq {1\over b^*(x,y)}.\eqno(3.3)$$ 
Moreover, 
$$\delta_M(x;y)={1\over b^*(x,y)}.\eqno(3.4)$$ \par

\noindent {\bf Proof}. Fix $x\in K^o, \ y\in \RR^N$, and $b<b^*(x,y)$. For $p$ a polynomial 
with $||p||_K\leq 1$ and $|p(x)|\not = 1$, 
let $T(t):=p(r(t))$ where $r(t)$ is as in (3.1). Then $T(t)$ is a trigonometric polynomial 
with deg$T=$deg$p$ and $||T||_{[0,2\pi]}\leq ||p||_K$ since 
$E_b(x,y)\subset K$. By  Szeg\"o's inequality for trigonometric polynomials, 
$${|T'(t)|\over \sqrt {1-T(t)^2}}\leq {\rm deg}T,$$
so that, in particular,
$$ {1\over {\rm deg}T}{|T'(0)|\over \sqrt {1-T(0)^2}}\leq 1.$$
But $T(0)=p(r(0))=p(x)$, $T'(0)=\nabla p(r(0))\cdot r'(0)=\nabla p(x)\cdot by=bD_yp(x)$, thus
$${1\over {\rm deg}p}{|D_yp(x)|\over \sqrt {1-(p(x))^2}}\leq 1/b$$
which gives (3.3). 

To show $\delta_M(x;y) \geq {1\over b^*(x,y)}$, by definition of $b^*(x,y)$, there exists $u\in \partial K\cap E_{b^*}(x,y)$; 
by symmetry, $-u\in \partial K\cap
E_{b^*}(x,y)$ as well. Let $H$ and $-H$ be support hyperplanes to $\partial K$ at $u,-u$ and let 
$n$ be a unit normal vector for $H$ (oriented ``out'' of $K$). 
Define the half-space
$$H_u:=\{z \in \RR^N:n\cdot z \leq n\cdot u\}.$$
Then $K\subset H_u\cap -(H_u)$ and hence
$$E_{b^*}(x,y)\subset K \subset H_u\cap -(H_u).$$
Let $p(z):={n\cdot z \over n\cdot u}$. By construction, $||p||_K\leq 1$ and $p$ maps $E_{b^*}(x,y)$ onto $[-1,1]$. Hence, with $r(t)=x\cos t+yb^*(x,y)\sin t$, we
can write
$$p(r(t))=A \cos t + B \sin t$$
for some $A,B$ with $A^2+B^2=1$. For if $A^2+B^2>1$, $p(E_{b^*}(x,y))\not \subset [-1,1]$; if $A^2+B^2<1$, $p(E_{b^*}(x,y))\subset [-1,1]$ but $p(E_{b^*}(x,y))\not
= [-1,1]$. Using the facts that deg$p=1; \ r(0)=x;$ and  $r'(0)=yb^*(x,y)$, it follows that
$$\delta_M(x;y)\geq {|D_yp(x)|\over \sqrt {1-(p(x))^2}}={1\over b^*(x,y)}\bigl ({|{d\over dt}(p(r(t)))|\over \sqrt
{1-(p(r(t)))^2}}|\bigr )_{t=0}$$
$$={1\over b^*(x,y)}\bigl ({|{d\over dt}(A\cos t + B \sin t)|\over \sqrt {1-(A\cos t + B \sin t))^2}}|\bigr )_{t=0}$$
$$={1\over b^*(x,y)}{|B|\over \sqrt {1-A^2}}={1\over b^*(x,y)}{|B|\over |B|}={1\over b^*(x,y)}$$
provided $A\not =1$. But $A=p(x)\not =1$ since $x\in K^o$. \hfill
$\clubsuit$ \par
\vskip6pt

\noindent In [4], it was shown that the equality 
$$\delta_B(x;y)={1\over b^*(x,y)}$$
holds for general convex bodies in $\RR^n$, and, moreover, the function $\delta_B$ is continuous.

\proclaim Corollary 3.3. \ Let $K$ be centrally symmetric and convex. Then $\delta_M=\delta_M^{(1)}$ where
$$\delta_M^{(1)}(x;y):=\sup_{||p||_K\leq 1, \ {\rm deg}p= 1}{|D_yp(x)|\over \sqrt {1-p(x)^2}}.$$
\par

\noindent {\bf Proof}. This follows since the proof of Lemma 3.2  shows that the supremum in the definition of $\delta_M(x;y)$ is attained 
for linear polynomials.  \hfill
$\clubsuit$ \par
\vskip6pt

We next show that the Dubiner distance $d_D$ is a $C^1$ pseudodistance (recall Definition 1.3 and equation (2.2)).

\proclaim Proposition 3.4. \ Let $K$ be centrally symmetric and convex. For all $E\subset \subset K^o$ and all $\epsilon >0$, there exists $\eta >0$ with
$$|d_D(a,b)-\delta_M(x;b-a)|\leq \epsilon |b-a|$$
for all $a,b\in B(x,\eta)$ and $x\in E$. \par

\noindent {\bf Proof}. Fix a positive integer $n$ and a polynomial $p$ of degree at most $n$ with $||p||_K\leq 1$. 

\noindent {\sl Claim: For all $\epsilon$ there exists $\eta >0$ (depending on $n,E$ but not $p$) with}
$$\big |{1\over {\rm deg}p}|\cos^{-1}(p(b))-\cos^{-1}(p(a))|-{1\over {\rm deg}p}{|D_{b-a}(p(x))|\over \sqrt {1-p(x)^2}}\big |\leq \epsilon |b-a|$$
for all $a,b\in B(x,\eta)$ and $x\in E$.

\noindent {\sl Proof of Claim}: Let $f(x)=\cos ^{-1}(p(x))$. It suffices to show that
$$\big |{|f(b)-f(a)|\over |b-a|}-|D_{b-a\over |b-a|}f(x)|\big |\leq \epsilon$$
for all $a,b\in B(x,\eta)$ and $x\in E$. To verify this, let $g(t):=f(a+t(b-a))$. Then
$$|g(1)-g(0)|=|f(b)-f(a)|=|\int_0^1g'(t)dt|$$
$$=\bigl |\int_0^1 D_{b-a\over |b-a|}f(a+t(b-a))dt\bigr | |b-a|$$
so that
$$\big |{|f(b)-f(a)|\over |b-a|}-|D_{b-a\over |b-a|}f(x)|\big |\leq \int_0^1 |D_{b-a\over |b-a|}f(a+t(b-a))-D_{b-a\over |b-a|}f(x)|dt$$
$$\leq \sup_{z\in [a,b]}|D_{b-a\over |b-a|}f(z)-D_{b-a\over |b-a|}f(x)|\leq \sup_{z\in B(x,\eta)}|D_{b-a\over |b-a|}f(z)-D_{b-a\over |b-a|}f(x)|.$$
This last quantity is less than $\epsilon$ if $\eta =\eta (n,E)$ is sufficiently small by compactness of the family $\{p:{\rm deg}p\leq n, \ ||p||_K\leq 1\}$.
This proves the claim.

>From Corollary 3.3, $\delta_M=\delta_M^{(1)}$, thus for $x\in E$ and $a,b\in B(x,\eta)$ we can take $p$ with deg$p=1$ and $||p||_K\leq 1$ such that
${|D_{b-a}(p(x))|\over \sqrt {1-p(x)^2}}>\delta_M(x;b-a)-\epsilon|b-a|$. Applying the Claim (with $n=1$), 
$$\delta_M(x;b-a) <2\epsilon|b-a| +|\cos^{-1}(p(b))-\cos^{-1}(p(a))|$$
$$\leq 2\epsilon|b-a| +d_D(a,b).$$
On the other hand, from Lemma 2.2, for $a,b\in B(x,\eta)$ and $x\in E$ we can find an $n$ with $d_D(a,b)=d_D^{(n)}(a,b)$. Choose 
$p$ with deg$p\leq n$ and $||p||_K\leq 1$ such that ${1\over {\rm deg}p}|\cos^{-1}(p(b))-\cos^{-1}(p(a))| >d_D(a,b)-\epsilon|b-a|$. Applying the Claim,
$$d_D(a,b) <2\epsilon|b-a| +{1\over {\rm deg}p}{|D_{b-a}(p(x))|\over \sqrt {1-p(x)^2}}\leq 2\epsilon|b-a| +\delta_M(x;b-a).$$
 \hfill
$\clubsuit$ \par
\vskip6pt

\proclaim Corollary 3.5. \ Let $K$ be centrally symmetric and convex. Then $\delta_M=Dd_D$ is continuous and $d_D$ is $C^1$. 
\par

\noindent {\bf Proof}. We have $\delta_M=Dd_D$ by (2.2) for general $K$. Thus $\delta_M$ is usc. By Corollary 3.3, $\delta_M$ is the supremum of a family of
continuous functions and hence is lowersemicontinuous (lsc). The fact that $d_D$ is a $C^1$ pseudodistance now follows from Proposition 3.4 and Definition 1.3. 
\hfill
$\clubsuit$ \par
\vskip6pt

>From Baran's work [1], for $K$ centrally symmetric and convex,
$$\delta_B(x;y)=\sup \{{|y\cdot w|\over \sqrt {1-(x\cdot w)^2}}: w\in K^*\} \eqno (3.5)$$
where recall
$$K^*:=\{x\in \RR^N:x\cdot y\leq 1 \ \hbox{for all} \ y \in K\}$$
is the polar of $K$. Note also that $|||x|||_K=\sup \{x\cdot w:w\in K^*\}$.

\proclaim Proposition 3.6. \ Let $K$ be centrally symmetric and convex. Then
$$b^*(x;y)=\inf \{ {\sqrt {1-(x\cdot w)^2}\over |y\cdot w|}: w\in K^*\}. \eqno(3.6)$$
Hence 
$$\delta_D=\delta_M=\delta_B. \eqno(3.7)$$
Moreover, 
$$d_D^i=d_M=d_B. \eqno(3.8)$$
\par

\noindent {\bf Proof}. From the definition of $b^*(x;y)$ in (3.2) and $K^*$ we can write
$$b^*(x;y)=\sup \{b: \sup_{w\in K^*, \ t\in [0,2\pi]} |x\cos t\cdot w +yb\sin t\cdot w|=1\}$$
$$=\sup \{b: \sup_{w\in K^*} [(w\cdot x)^2+b^2(w\cdot y)^2]= 1\}.$$
To see that this last supremum equals $\inf \{ {\sqrt {1-(x\cdot w)^2}\over |y\cdot w|}: w\in K^*\}$, take any $b$ with $\sup_{w\in K^*} [(w\cdot x)^2+b^2(w\cdot
y)^2]= 1$; then, for any $w\in K^*$, $(w\cdot x)^2+b^2(w\cdot y)^2\leq 1$ so that $b\leq {\sqrt {1-(x\cdot w)^2}\over |y\cdot w|}$ which shows that $b^*$ is less
than or equal to the right-hand-side of (3.6). Next, we observe that the infimum in the right-hand-side of (3.6) is attained. Let $b_0 =\min \{ {\sqrt {1-(x\cdot
w)^2}\over |y\cdot w|}: w\in K^*\}$. Then $b_0\leq {\sqrt {1-(x\cdot w)^2}\over |y\cdot w|}$ for all $w\in K^*$ with equality at some point(s); hence
$(w\cdot x)^2+b_0^2(w\cdot y)^2 \leq1$ for all $w\in K^*$ with equality at some point(s); i.e., $b_0\leq b^*$ and equality holds. 

Equation (3.7) follows from 
(2.2), (3.4), (3.5) and (3.6). Using this, Proposition 1.6 gives  
$$d_D^i=\int \delta_D= \int \delta_M =\int \delta_B$$
which is (3.8).
 \hfill
$\clubsuit$ \par
\vskip6pt

As a concrete example, for $K$ the closed unit ball in $\RR^N$, given $x\in K^o$ and $y\in \RR^N$, let
$$\tilde w :=y(1-|x|^2)+(y\cdot x)x.$$
Then $w:=\tilde w/|\tilde w|$ maximizes ${|y\cdot w|\over \sqrt {1-(x\cdot w)^2}}$ and this maximal value is
$$\bigl ({(1-|x|^2)|y|^2+(x\cdot y)^2\over 1-|x|^2}\bigr )^{1/2}\delta_B(x;y).$$

We conjecture that $$d_D(a,b)=d_M(a,b)=d_B(a,b) \eqno(3.9)$$ for $K$ centrally symmetric and convex.  
We present some evidence supporting the validity of the conjecture.

\proclaim Proposition 3.7. \ For a centrally symmetric 
$E\subset \RR^N$ bounded by an ellipsoid, $d_D^{(1)}=d_D=d_M=d_B$. 

\par

\noindent {\bf Proof}. We have equality of $d_D, \ d_D^{(1)}$ and $d_B$ for $K=\overline B$, the unit ball,  
by explicit calculation in [3]. Thus by inequality (2.1), 
$$d_D^{(1)}=d_D=d_M=d_B \eqno(3.10)$$
for $K=\overline B$. Since $E=T(K)$ for some invertible linear mapping
$T$, equality holds in (3.10) for $E$ from (2.9) and the observation that $d_D^{(1)}$ for $K$ and $E$ coincide.  \hfill $\clubsuit$ \par
\vskip6pt

We now specialize to centrally symmetric convex bodies in $\RR^2$.

\proclaim Theorem 3.8. \ Let $K\subset \RR^2$ be centrally symmetric and convex. For two points $a,b\in K$ with the property that 
there exists a centrally symmetric region $E=E(a,b)\subset K$ bounded by an ellipse with $a,b$ lying on the same `side'  of 
the ellipse $\partial E$ (with `sides' separated by an axis joining supporting hyperplanes), we have 
$d_D^{(1)}(a,b)=d_D(a,b)=d_M(a,b)=d_B(a,b)$.
\par

\noindent {\bf Proof}. The idea is similar to that utilized in Lemma 3.2. We expand $E$ to construct a centrally symmetric region $\tilde E \subset K$ bounded by an
ellipse with
$a,b\in
\partial
\tilde E$ with the property that there exists
$u\in \partial K \cap \partial \tilde E$, and hence $-u\in \partial K \cap \partial \tilde E$. (cf., Theorem 5.3 and its proof in [6]). Then, letting $H,-H$ be
support hyperplanes to
$\partial K$ at
$u,-u$ and calling 
$n$ the unit normal vector for $H$ (oriented ``out'' of $K$), the half-space $H_u:=\{z:n\cdot z \leq n\cdot u\}$ satisfies $K\subset H_u\cap -(H_u)$ and hence
$$\tilde E\subset K \subset H_u\cap -(H_u).$$
Let $p(z):={n\cdot z \over n\cdot u}$. By construction, $||p||_K\leq 1$ and $p$ maps $\tilde E$ and $K$ onto $[-1,1]$. Thus
$$d_D(a,b)=d_D^K(a,b)\geq |\cos^{-1}(p(b))-\cos^{-1}(p(a))|$$
$$=d_D^{\tilde E}(a,b)=d_M^{\tilde E}(a,b)=d_B^{\tilde E}(a,b),$$
the last line coming from Proposition 3.7. 
But from Remark 2.1, (recall that $d_D^{\tilde E}$ is well-definded on $\partial {\tilde E}$)
we have $d^K(a,b) \leq d^{\tilde E}(a,b)$ on $\tilde E \times \tilde E$ for each of our three distances and the result follows. \hfill
$\clubsuit$
\par 
\vskip6pt

>From the proof of Theorem 3.8, we see that equality holds in (3.9) at points $(a,b)\in E_{b^*}(x,y)\times E_{b^*}(x,y)$ for each centrally symmetric ellipse
$E_{b^*}(x,y)$ contained in $K$ with $b^*(x;y)$ as in (3.6). Now recall from Remark 1.4 that the Markov and Baran distances are always inner; i.e.,
$d_M=d_M^i$ and
$d_B=d_B^i$. Suppose we knew that the Dubiner distance
$d_D$ on a centrally symmetric convex body was an inner distance. Then from (3.8) of Proposition 3.6 
we conclude that (3.9) holds. In this vein, we mention the following definition. For a subset $X$ of a vector space 
equipped with a distance $d$, the pair $(X,d)$ is called {\it metrically convex} if given $a,b\in X$, there exists $c\in X$ with
$d(a,c)+d(c,b)=d(a,b)$. It is known [11] that {\it if $(X,d)$ is metrically convex and complete, then through each pair of points $a,b$ in $X$ there is a
shortest curve; i.e. there exists $\alpha:[0,1]\to X$ a continuous curve joining $a$ and $b$ with $L_d(\alpha)=d^i(a,b)$, and, indeed, $d=d^i$}. Thus we make the
following observation.

\proclaim Corollary 3.9. \ Let $K\subset \RR^2$ be centrally symmetric and convex with the additional property that for any two points $a,b\in K$, there exists
a centrally symmetric region $E=E(a,b)\subset K$ bounded by an ellipse with $a,b\in \partial E$. Then $d_D=d_M=d_B$. \par

\noindent {\bf Proof}. The property that $d_D$ locally dominates the Euclidean distance in the interior of $K$ (see Remark
1.3) extends to $K$, implying completeness of $(K,d_D)$. Therefore it 
suffices to show that
$d_D=d_D^i$ which will follow if
$d_D$ is metrically convex. But this follows by the hypothesized property, since we can take $c$ to be any
point on the (shorter) arc of the ellipse
$\partial \tilde E$ joining $a$ and $b$ which was constructed in the proof of Theorem 3.8. \hfill $\clubsuit$ \par
\vskip6pt

The geometric property hypothesized in Corollary 3.9 does not hold for every centrally symmetric convex body $K\subset \RR^2$. For example, take $K$ to be the
square
$[-a,a]\times [-a,a]$ (it can be shown, however, that a square is, indeed, metrically convex). By rounding off the edges of the square, we can even construct such a $K$ which is strictly convex with smooth boundary.





\vfill \eject

 


%\vskip10pt

\References
\ref Baran, M., Bernstein type theorems for compact sets in $\RR^n,$ \JAT~{\bf 69},
(1992), 156--166.

\ref Baran, M., Bernstein type theorems for compact sets in $\RR^n$ revisited, \JAT~{\bf 79},
(1994), 190--198.

\ref Bos, L., Levenberg, N.,  and Waldron, S., Metrics associated to multivariate polynomial inequalities, 
{\sl Advances in Constructive Approximation}, 
M. Neamtu and E. B. Saff eds., Nashboro Press, Nashville, 2004, 133--147.

\ref Burns, D., Levenberg, N.,  Ma'u, S. and R\'ev\'esz, Sz.,  
Monge-Amp\`ere Measures for Convex Bodies and Bernstein-Markov
Type Inequalities, submitted.

\ref Dineen, S., {\sl The Schwarz Lemma}, Clarendon Press, Oxford, 1989.  

\ref Dubiner, M., The theory of multidimensional polynomial approximation,
J. Anal. Math. {\bf 67}, (1995), 39--116.

\ref Jarnicki, M.  and Pflug, P., {\sl Invariant Distances and Metrics
in Complex Analysis}, de Gruyter, Berlin, 1993.

\ref Klimek, M., {\sl Pluripotential Theory}, Clarendon Press, Oxford, 1991.

\ref Kro\'o, A. and  R\'ev\'esz, Sz., {\sl On Bernstein and Markov-type inequalities for
multivariate polynomials on convex bodies}, \JAT~{\bf 99} (1999), 134--152.

\ref Milev, L. and S. R\'ev\'esz, Bernstein's inequality for multivariate polynomials 
on the standard simplex, J. Ineq. Appl. 2005:2 (2005), 145--163.

\ref Rinow, W., {\sl Die Innere Geometrie der Metrischen R\"aume}, Springer-Verlag, 1961.

\ref Sarantoupolos, {\sl Bounds on the derivatives of polynomials on Banach spaces}, Math. Proc. Cambridge Philos. 
Soc. {\bf 110} (1991), 307--312.

%******************************* ADDRESS ************************* 

{%  Beginning of addresses

\bigskip\obeylines
Len Bos
Department of Mathematics and Statistics
University of Calgary
Calgary, Alberta
Canada T2N 1N4
{\tt lpbos@math.ucalgary.ca}

\bigskip
Norm Levenberg
Department of Mathematics
Indiana University 
Bloomington, IN 47405
USA
{\tt nlevenbe@indiana.edu}

\bigskip
Shayne Waldron
Department of Mathematics
University of Auckland
Private Bag 92019
Auckland, New Zealand
{\tt waldron@math.auckland.ac.nz}
}  %End of addresses

\bye  