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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: March 21, 2000
MathSciNet review: 1751449
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Abstract:

For a given convex (semi-convex) function $u$, defined on a nonempty open convex set $\Omega\subset\mathbf{R}^n$, 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 $r\in \{0,\ldots,n\}$, the $r$-th coefficient measure of the local Steiner formula for $u$, restricted to the set of $r$-singular points of $u$, is absolutely continuous with respect to the $r$-dimensional Hausdorff measure, and that its density is the $(n-r)$-dimensional Hausdorff measure of the subgradient of $u$.

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


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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: http://dx.doi.org/10.1090/S0002-9947-00-02671-4
PII: S 0002-9947(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