Affine vs. Euclidean isoperimetric inequalities

TL;DR

This paper explores the relationship between affine and Euclidean isoperimetric inequalities, introducing a family of inequalities derived from measures on the sphere, with the Petty projection inequality being uniquely affine invariant.

Contribution

It establishes a family of isoperimetric inequalities from spherical measures and identifies the Petty projection inequality as the only affine invariant among them.

Findings

The Petty projection inequality is the only affine invariant among the derived inequalities.

A family of sharp Sobolev inequalities for functions of bounded variation is obtained.

New Lp isoperimetric and Sobolev inequalities are established.

Abstract

It is shown that every even, zonal measure on the Euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical Euclidean isoperimetric inequality. The strongest member of this large family of inequalities is shown to be the only affine invariant one among them - the Petty projection inequality. As an application, a family of sharp Sobolev inequalities for functions of bounded variation is obtained, each of which is stronger than the classical Sobolev inequality. Moreover, corresponding families of Lp isoperimetric and Sobolev type inequalities are also established.