# Maximum of a function on $d-$dimensional convex compact sets

Let $\mathcal C_d$ denote the set of all $d-$dimensional convex compact subsets with barycenter at the origin of the $d$-dimensional Euclidean space $\mathbb E^d$. Given an element $C\in\mathcal C_d$ with boundary $\partial C$, define $\Sigma_0\subset \partial C$ by $\Sigma_0=\partial C\cap(-\partial C)$. The complement $\partial C\setminus \Sigma_0$ is open and can be partitioned into two open subsets $\Sigma_\pm$ defined by $$\Sigma_-=\lbrace P\in\partial C\vert -P\in\mathrm{Int}(C)\rbrace$$ and $$\Sigma_+=\lbrace P\in \partial C\vert -P\not\in C\rbrace\ .$$ Set $\rho(C)=\mathrm{Area}(\Sigma_+)/\mathrm{Area}(\Sigma_-)$ (using the convention $\rho(C)=1$ if $\Sigma_+=\Sigma_-=\emptyset$ which happens exactly if $C$ is centrally symmetric) where $\mathrm{Area}(\Sigma_\pm)$ is the $(d-1)$ dimensional area.

The function $\rho:\mathcal C_d\longrightarrow [1,\infty)$ is invariant under the action of the linear group on $\mathcal C_d$ and is bounded. What is its maximal value (and on which convex sets is it achieved)?

Added: My argument for the invariance under the linear group is flawed. I am also no longer sure that $\rho$ is bounded (my proof used invariance under the linear group).

(Added after a moments thought: One remedy is to replace $\rho(C)$ by $\mathrm{Vol}(C)/\mathrm{Vol}(C\cap(-C))$ which is obviously invariant under linear transformations and which is bounded.)

A probably naive guess for the maximum is the value of $\rho$ on a simplex. In dimension $2$, one gets $\rho(\Delta)=2$ if $\Delta\in\mathcal C_2$ is a triangle and in dimension $3$ one gets $\rho(\Delta)=3$ for $\Delta$ a simplex in $\mathcal C_3$. I ignore the value of $\rho$ on simplices of dimension $\geq 4$ (question Name of a polytope is motivated by the computation of these values).

A similar invariant which has perhaps been studied is obtained by considering $\lambda_C=\max_L_+\mathrm{length}(L_+\cap C)/\mathrm{length}((-L_+)\cap C)$ where the maximum is over the set of all half-lines rooted at the origin. The invariant $\lambda(C)$ is also bounded and invariant under the action of the linear group on $\mathcal C_d$. (Here I am sure since this is obvious.)

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You probably have already seen this, but for anyone else happening across this question, you probably want to look for papers about John's ellipsoids. In particular, look for papers by Keith Ball. I took a course from him a few years ago about this stuff which I don't remember as much of as I would like. However, one thing we did do was show that the simplex was, in some rigorous sense, the "least centrally symmetric" convex body, which agrees with your naive assessment. –  Aubrey da Cunha Mar 31 '11 at 18:18