## Duke Mathematical Journal

### On the Oberlin affine curvature condition

Philip T. Gressman

#### Abstract

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 equiaffine-invariant measure exists and that, modulo sufficient regularity assumptions on the submanifold, the measure satisfies the affine curvature condition of 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 concerning inequalities of reverse Sobolev type for polynomials on arbitrary measurable subsets of the real line.

#### Article information

Source
Duke Math. J., Volume 168, Number 11 (2019), 2075-2126.

Dates
Revised: 16 October 2018
First available in Project Euclid: 3 July 2019

https://projecteuclid.org/euclid.dmj/1562119267

Digital Object Identifier
doi:10.1215/00127094-2019-0010

Mathematical Reviews number (MathSciNet)
MR3992033

#### Citation

Gressman, Philip T. On the Oberlin affine curvature condition. Duke Math. J. 168 (2019), no. 11, 2075--2126. doi:10.1215/00127094-2019-0010. https://projecteuclid.org/euclid.dmj/1562119267

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