## Question:

Let's suppose that $S \subset \mathbb{R}^n$ is convex and symmetric so:

\begin{equation} x \in S \iff -x \in S \tag{1} \end{equation}

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

\begin{equation} R = \sup_{x \in S} \lVert x \rVert \tag{2} \end{equation}

and use (2) to define:

\begin{equation} V = \{x \in S: \lVert x \rVert = R\} \tag{3} \end{equation}

then I conjecture that:

\begin{equation} S = \text{conv}(V) \tag{*} \end{equation}

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:

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.

I was thinking in particular about symmetric convex polytopes and my intuition was that all symmetric convex polytopes in $\mathbb{R}^n$ 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.