Steiner type formulae and weighted measures of singularities for semi-convex functions

Authors:
Andrea Colesanti and Daniel Hug

Journal:
Trans. Amer. Math. Soc. **352** (2000), 3239-3263

MSC (2000):
Primary 26B25, 52A41; Secondary 28A78, 52A20, 49J52, 49Q15

DOI:
https://doi.org/10.1090/S0002-9947-00-02671-4

Published electronically:
March 21, 2000

MathSciNet review:
1751449

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Abstract | References | Similar Articles | Additional Information

For a given convex (semi-convex) function , defined on a nonempty open convex set , we establish a local Steiner type formula, the coefficients of which are nonnegative (signed) Borel measures. We also determine explicit integral representations for these coefficient measures, which are similar to the integral representations for the curvature measures of convex bodies (and, more generally, of sets with positive reach). We prove that, for , the -th coefficient measure of the local Steiner formula for , restricted to the set of -singular points of , is absolutely continuous with respect to the -dimensional Hausdorff measure, and that its density is the -dimensional Hausdorff measure of the subgradient of .

As an application, under the assumptions that is convex and Lipschitz, and is bounded, we get sharp estimates for certain weighted Hausdorff measures of the sets of -singular points of . Such estimates depend on the Lipschitz constant of and on the quermassintegrals of the topological closure of .

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Additional Information

**Andrea Colesanti**

Affiliation:
Universitá Degli Studi di Firenze, Dipartimento di Matematica “U. Dini”, Viale Morgagni 67/A, 50134 Firenze, Italy

Email:
colesant@udini.math.unifi.it

**Daniel Hug**

Affiliation:
Mathematisches Institut, Albert-Ludwigs-Universität, Eckerstraße 1, D-79104 Freiburg i. Br., Germany

Email:
hug@sun1.mathematik.uni-freiburg.de

DOI:
https://doi.org/10.1090/S0002-9947-00-02671-4

Keywords:
Steiner formula,
convex function,
semi-convex function,
singularities,
weighted Hausdorff measures,
subgradient map,
unit normal bundle,
non-smooth analysis

Received by editor(s):
December 30, 1996

Published electronically:
March 21, 2000

Article copyright:
© Copyright 2000
American Mathematical Society