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15 August 2019 On the Oberlin affine curvature condition
Philip T. Gressman
Duke Math. J. 168(11): 2075-2126 (15 August 2019). DOI: 10.1215/00127094-2019-0010

Abstract

We generalize the well-known notions of affine arclength and affine hypersurface measure to submanifolds of any dimension d in Rn, 1dn1. 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.

Citation

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Philip T. Gressman. "On the Oberlin affine curvature condition." Duke Math. J. 168 (11) 2075 - 2126, 15 August 2019. https://doi.org/10.1215/00127094-2019-0010

Information

Received: 28 November 2017; Revised: 16 October 2018; Published: 15 August 2019
First available in Project Euclid: 3 July 2019

zbMATH: 07114914
MathSciNet: MR3992033
Digital Object Identifier: 10.1215/00127094-2019-0010

Subjects:
Primary: 44A12
Secondary: 42B05, 53A15

Rights: Copyright © 2019 Duke University Press

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Vol.168 • No. 11 • 15 August 2019
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