Mean inequalities for symmetrizations of convex sets

Speaker: 

Katherina von Dichter

Institution: 

Technische Universität München

Time: 

Monday, September 12, 2022 - 12:00pm

Location: 

Zoom

The arithmetic-harmonic mean inequality can be generalized for convex sets, considering the intersection, the harmonic and the arithmetic mean, as well as the convex hull of two convex sets. We study those relations of symmetrization of convex sets, i.e., dealing with the means of some convex set C and -C. We determine the dilatation factors, depending on the asymmetry of C, to reverse the containments between any of those symmetrizations, and tighten the relations proven by Firey and show a stability result concerning those factors near the simplex.

https://sites.google.com/view/paw-seminar/

On the Fourier-Entropy Conjecture

Speaker: 

Dor Minzer

Institution: 

MIT

Time: 

Monday, April 4, 2022 - 12:00pm

Location: 

Zoom: https://sites.google.com/view/paw-seminar

Characterizing Boolean functions with small total influence is one of the most fundamental questions in analysis of Boolean functions. 

The seminal results of Kahn-Kalai-Linial and of Friedgut address this question for total influence $K = o(\log n)$, and show that 

a function with total influence $K$ (essentially) depends on $2^{O(K)}$ variables. 

The Fourier-Entropy Conjecture of Friedgut and Kalai is an outstanding conjecture that strengthens these results, and remains

meaningful for $K \geq \log n$. Informally, the conjecture states that the Fourier transform of a function with total influence $K$, 

is concentrated on at most $2^{O(K)}$ distinct characters. 

In this talk, we will discuss recent progress towards this conjecture. We show that functions with total influence $K$ are concentrated 

on at most $2^{O(K\log K)}$ distinct Fourier coefficients. We also mention some applications to learning theory and sharp thresholds.

Based on a joint work with Esty Kelman, Guy Kindler, Noam Lifshitz and Muli Safra.

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