We say a subset $A$ of $\mathbb{R}^N$ is balanced if \begin{equation} x \in A, \lambda \in [-1,1] \implies \lambda x \in A. \end{equation} Given a subset $A$ of $\mathbb{R}^N$, we write \begin{equation} A^{\oplus k} := \{ x_1 + \ldots + x_k : x_i \in A \} \end{equation} for the $k$-fold Minkowski sum of $A$ with itself. Note that if $A$ balanced, so is $A^{\oplus k}$.
Talagrand has conjectured the following:
There is a universal integer $k$ such that for all $N$, for all balanced subsets $A$ of $\mathbb{R}^N$ of standard Gaussian measure at least $1/2$, $A^{\oplus k}$ contains a convex subset of Gaussian measure at least $1/10$.
This conjecture is made on Talagrand's webpage here: https://michel.talagrand.net/prizes/convexity.pdf
My question: What is currently known about this conjecture?
While this conjecture seems like an absolutely fundamental conjecture in convex geometry, I have been able to find only very little literature discussing it explicitly. And Talagrand promises 1000 dollars to anyone with a solution! :)
A few further remarks.
First, the requirement that $A$ is balanced may not be important. The constants $1/2$ and $1/10$ may not be important either. The Gaussian measure (as opposed to e.g. uniform measure on the unit cube, or any product measure) may not be important either.
Also, the conjecture may be posed in terms of the least universal $k(N)$ such that $k(N)$ Minkowski additions are required to create convexity in dimension $N$. By Carathéodory's theorem we have $k(N) \leq N+1$. Below is a reference where Talagrand proves $k(N) \geq 3$ for large $N$.
Talagrand discusses this conjecture in a few places. This conjecture and several others are posed in:
Talagrand, Michel, Are many small sets explicitly small?, Proceedings of the 42nd annual ACM symposium on theory of computing, STOC ’10. Cambridge, MA, USA, June 5–8, 2010. New York, NY: Association for Computing Machinery (ACM) (ISBN 978-1-60558-817-9). 13-36 (2010). ZBL1293.60014.
The follow article Talagrand uses a quite intricate argument to say that the conjecture is certainly false for $k=2$, so $k$ needs to be at least $3$. The argument used seems to break down completely for $k=3$.
Talagrand, M., Are all sets of positive measure essentially convex?, Lindenstrauss, J. (ed.) et al., Geometric aspects of functional analysis. Israel seminar (GAFA) 1992-94. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 77, 294-310 (1995). ZBL0835.60003.
I should also note that Frankston, Kahn, Narayanan and Park prove one of the conjectures in the former reference above ("Are many small sets explicitly small") in this paper:
Frankston, Keith; Kahn, Jeff; Narayanan, Bhargav; Park, Jinyoung, Thresholds versus fractional expectation-thresholds, Ann. Math. (2) 194, No. 2, 475-495 (2021). ZBL1472.05132.
but I can't quite see the link to the convex geometry conjecture above.
Thank you for reading, and thanks in advance for any comments!
