4
$\begingroup$

A is a closed convex set with non-empty interior. Does A must equal to the convex hull of its boundary?

I know this is false when A is half space. But what about other sets?

$\endgroup$

1 Answer 1

6
$\begingroup$

Let $A\subseteq \mathbb{R}^d$ be a convex and closed set with non-empty interior. Then $A$ does not coincide with the convex hull of its boundary if and only if it is either a half-space or $\mathbb{R}^d$ itself. Source: Lemma 1.4.1 in R. Schneider: Convex Bodies: The Brunn--Minkowski Theory, Cambridge University Press, Cambridge (1993). It is a consequence of the separation theorem.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.