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Lemma1
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Let $X\subseteq \mathbb{R}^n$ be an open set and let $Y\subseteq X$ be convex. Can I always construct an open and convex set $Z$ such that $Y\subseteq Z\subseteq X$?

Edit: As Ilya Bogdanov pointed out, the claim is wrong in general. But, what if $X$ is the interior of a finite union of polytopes? Any hint will be greatly appreciated.

Let $X\subseteq \mathbb{R}^n$ be an open set and let $Y\subseteq X$ be convex. Can I always construct an open and convex set $Z$ such that $Y\subseteq Z\subseteq X$?

Let $X\subseteq \mathbb{R}^n$ be an open set and let $Y\subseteq X$ be convex. Can I always construct an open and convex set $Z$ such that $Y\subseteq Z\subseteq X$?

Edit: As Ilya Bogdanov pointed out, the claim is wrong in general. But, what if $X$ is the interior of a finite union of polytopes? Any hint will be greatly appreciated.

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Lemma1
Lemma1
  • 157
  • 4

Convex subsets of an open set

Let $X\subseteq \mathbb{R}^n$ be an open set and let $Y\subseteq X$ be convex. Can I always construct an open and convex set $Z$ such that $Y\subseteq Z\subseteq X$?