On the Oberlin affine curvature condition

TL;DR

This paper extends affine curvature measures to submanifolds of any dimension in Euclidean space, establishing a canonical affine invariant measure that satisfies the optimal affine curvature condition under regularity assumptions.

Contribution

It introduces a generalized affine invariant measure for submanifolds of arbitrary dimension and proves it satisfies the affine curvature condition with the best possible exponent.

Findings

Existence of a canonical affine invariant measure for submanifolds.

Verification that the measure satisfies the affine curvature condition with optimal exponent.

Development of new inequalities related to reverse Sobolev type for polynomials.

Abstract

In this paper we generalize the well-known notions of affine arclength and affine hypersurface measure to submanifolds of any dimension $d$ in $\mathbb R^n$ , $1 \leq d \leq n-1$. We show that a canonical affine invariant measure exists and that, modulo sufficient regularity assumptions on the submanifold, the measure satisfies the affine curvature condition of D. Oberlin with an exponent which is best possible. The proof combines aspects of Geometric Invariant Theory, convex geometry, and frame theory. A significant new element of the proof is a generalization to higher dimensions of an earlier result of the author concerning inequalities of reverse Sobolev type for polynomials on arbitrary measurable subsets of the real line.