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.