An inequality for functions on the Hamming cube

TL;DR

This paper establishes a new inequality for functions on the Hamming cube, linking isoperimetric properties with random walk behaviors, and characterizes subcubes as extremal sets for mean first exit time.

Contribution

It introduces a novel inequality extending the edge-isoperimetric inequality and connects it to random walk properties on the cube.

Findings

Subcubes maximize mean first exit time among equal-sized subsets.

The new inequality generalizes classical isoperimetric inequalities.

Equivalence between the inequality and the extremal property of subcubes.

Abstract

We prove an inequality for functions on the discrete cube extending the edge-isoperimetric inequality for sets. This inequality turns out to be equivalent to the following claim about random walks on the cube: Subcubes maximize 'mean first exit time' among all subsets of the cube of the same cardinality.