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 Question:

Let's suppose that $S subset mathbbR^n$ is convex and symmetric so:

beginequation
x in S iff -x in S tag1
endequation

Now, if we define the radius of $S$ as $R$ such that:

beginequation
R = sup_x in S lVert x rVert tag2
endequation

and use (2) to define:

beginequation
V = x in S: lVert x rVert = R tag3
endequation

then I conjecture that:

beginequation
S = textconv(V) tag*
endequation

I have worked out special cases of this problem within the context of high-dimensional probability but I suspect that it's generally true.

Might there be a theorem which guarantees this result?

Special case:

1. As some people are voting to close this question I'd like to share my intuition about a special case as I think it might clarify my perspective.




2. I was thinking in particular about symmetric convex polytopes and my intuition was that all symmetric convex polytopes in $mathbbR^n$ whose vertex set equalled $V$ in (3) were regular polytopes.

 Remark:

I consulted several texts on convexity in high dimensions and couldn't find an answer to this question. For this reason I decided to ask the question here.

Consider the set
$$S:= (x,y) in mathbb R^2 : x^2+4y^2 leq 4 $$

This is convex and symmetric, and $R=2$.

But $V= (2,0), (-2,0) $ and $mboxconv(V)= (x, 0) : -2 leq x leq 2 neq S$.

Take any convex, symmetric, bounded set $T$ in $mathbbR^n$. Choose any point $pin mathbbR^n$ such that $|p|>sup_xin T|x|$ and let $S$ be the convex hull of $Tcup pm p$. This set is convex and symmetric, $V=pm p$, and the convex hull of $V$ is just the line segment connecting $p$ and $-p$. It is easy to construct counterexmples in this way, for example if the original set $T$ has non-empty interior.