On sharp isoperimetric inequalities on the hypercube

Abstract

We prove the sharp isoperimetric inequality $$\mathbb{E}{h}_A^{{\log }_2(3/2)}\ge \hat{I}¼(A{)}^{\ast }({\log }_2(1/\hat{I}¼(A{)}^{\ast }){)}^{{\log }_2(3/2)}$$ for all sets $A\subseteq \{0,1{\}}^n$, where $\hat{I}¼$ denotes the uniform probability measure, $\hat{I}¼(A{)}^{\ast }=\min \{\hat{I}¼(A),1-\hat{I}¼(A)\}$, $h_A$ 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 $\mathbb{E}{h}_A^2$ in terms of $\hat{I}¼(A)$ for all ${\hat{I}}^2\in [1/2,1]$, improving, and in some cases tightening, previously known results. In particular, we obtain the sharp inequality $\mathbb{E}{h}_A^{0.53}\ge 2\hat{I}¼(A)(1-\hat{I}¼(A))$ for all sets with $\hat{I}¼(A)\ge 1/2$, 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 $f:\{-1,1{\}}^n\to X$, $\Vert f{\Vert }_{\infty }\le 1$, and any $p\in [1,2]$ we have $$\Vert Df{\Vert }_p\gtrsim \frac{1}{q^{3/2}{C}_q(X)}\Vert f{\Vert }_2^{2/p}{\left(\log \frac{e\Vert f{\Vert }_2}{\Vert f{\Vert }_1}\right)}^{1/q},$$ where $\Vert Df{\Vert }_p^p=\mathbb{E}\Vert {\sum }_{1\le j\le n}{x}_j^{\prime }{D}_{j}f(x){\Vert }^p$, $x^{\prime }$ 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.

17 pages