Duke Mathematical Journal
- Duke Math. J.
- Volume 168, Number 11 (2019), 2075-2126.
On the Oberlin affine curvature condition
We generalize the well-known notions of affine arclength and affine hypersurface measure to submanifolds of any dimension in , . 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.
Duke Math. J., Volume 168, Number 11 (2019), 2075-2126.
Received: 28 November 2017
Revised: 16 October 2018
First available in Project Euclid: 3 July 2019
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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