On sharp isoperimetric inequalities on the hypercube

TL;DR

This paper establishes sharp isoperimetric inequalities on the hypercube, improves bounds for certain set measures, and extends Talagrand's inequalities to Banach space-valued functions, advancing understanding of combinatorial and functional inequalities.

Contribution

It proves a new sharp isoperimetric inequality for hypercube sets, refines bounds on expected neighbor counts, and extends Talagrand's inequalities to Banach spaces with finite cotype.

Findings

Sharp isoperimetric inequality for all subsets of the hypercube.

Improved bounds on expected neighbor counts for sets with measure ≥ 1/2.

Extension of Talagrand's inequalities to Banach space-valued functions.

Abstract

We prove the sharp isoperimetric inequality $$ \mathbb{E} \,h_{A}^{\log_{2}(3/2)} \geq \mu(A)^{*} (\log_{2}(1/\mu(A)^{*}))^{\log_{2}(3/2)} $$ for all sets $A \subseteq \{0,1\}^n$, where $\mu$ denotes the uniform probability measure, $\mu(A)^{*}=\min\{\mu(A), 1-\mu(A)\}$, $h_A$ is supported on $A$ and to each vertex $x$ assigns the number of neighbour vertices in the complement of $A$. The inequality becomes equality for any subcube. Moreover, we provide lower bounds on $\mathbb{E} h_{A}^{\beta}$ in terms of $\mu(A)$ for all $\beta \in [1/2,1]$, improving, and in some cases tightening, previously known results. In particular, we obtain the sharp inequality $\mathbb{E}h_{A}^{0.53}\geq 2 \mu(A)(1-\mu(A))$ for all sets with $\mu(A)\geq 1/2$, which allows us to refine a recent result of Kahn and Park on isoperimetric inequalities about partitioning the hypercube. Furthermore, we derive Talagrand's isoperimetric inequalities for functions with values in a Banach space having finite cotype: for all $f :\{-1,1\}^{n} \to X$, $\|f\|_{\infty}\leq 1$, and any $p \in [1,2]$ we have $$ \|Df\|_{p} \gtrsim \frac{1}{q^{3/2}C_{q}(X)} \|f\|_{2}^{2/p}\left(\log \frac{e\|f\|_{2}}{\|f\|_{1}}\right)^{1/q}, $$ where $\| Df\|_{p}^{p} = \mathbb{E} \| \sum_{1\leq j \leq n} x'_{j} D_{j} f(x)\|^{p}$, $x'$ is independent copy of $x$, and $C_{q}(X)$ is the cotype $q$ constant of $X$. Different proofs of the recently resolved Talagrand's conjecture will be presented.