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Posted online: 2024-03-21 15:39:31Z by SciLag Admin27

Cite as: P-240321.2

Create convexity in 3 (or 100?) steps only! Consider an integer $N$. Let us say that a compact subset $A$ of $\mathbb{R}^N$ is balanced if $x \in A$, $\lambda \in \mathbb{R}$, $|\lambda| \leq 1 \Rightarrow \lambda x \in A$. Let us denote by $\gamma_N$ the canonical Gaussian measure on $\mathbb{R}^N$.

**Problem.** Prove that there exists an integer $q$, such that for all $N$ and every compact balanced set $A$ of $\mathbb{R}^N$ such that $\gamma_N(A) \geq 1/2$, one can find a convex compact set $C \subset A + \cdots + A$ (with $q$ terms on the right) such that $\gamma_N(C) \geq 1/2$.

In words: finitely many steps, independently of dimension, suffice to create convexity. Even if the constant $L$ is large one cannot always find such a $C$ inside $L(A + A)$, but I do not know if one can find it inside $L(A+A+A)$. This sounds to me as a fundamental question. The silence of the convexity specialists about it has been so far deafening.

This problem is discussed in my paper *Are all sets of positive measure essentially convex? *Operator theory: Advances and applications. Vol. 77, Birkhäuser, 1995, p. 295-310. It is somewhat related to the *combinatorics problem* but you will get 2000 USD for solving both.

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Edited: (general update ) at 2024-03-21 18:38:48Z

Created at: 2024-03-21 15:39:31Z View this version

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