Hessian measures of semi-convex functions and applications to support measures of convex bodies |
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Authors: | Andrea Colesanti Daniel Hug |
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Institution: | Dipartimento di Matematica “U. Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy. e-mail: andrea.colesanti@bb.math.unifi.it, IT Mathematisches Institut, Albert-Ludwigs-Universit?t, Eckerstra?e 1,?79104 Freiburg, Germany. e-mail: hug@sun2.mathematik.uni-freiburg.de, DE
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Abstract: | This paper originates from the investigation of support measures of convex bodies (sets of positive reach), which form a
central subject in convex geometry and also represent an important tool in related fields. We show that these measures are
absolutely continuous with respect to Hausdorff measures of appropriate dimensions, and we determine the Radon-Nikodym derivatives
explicitly on sets of σ-finite Hausdorff measure. The results which we obtain in the setting of the theory of convex bodies
(sets of positive reach) are achieved as applications of various new results on Hessian measures of convex (semi-convex) functions.
Among these are a Crofton formula, results on the absolute continuity of Hessian measures, and a duality theorem which relates
the Hessian measures of a convex function to those of the conjugate function. In particular, it turns out that curvature and
surface area measures of a convex body K are the Hessian measures of special functions, namely the distance function and the support function of K.
Received: 15 July 1999 |
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Keywords: | Mathematics Subject Classification (1991): Primary 52A20 26B25 Secondary 53C65 28A78 |
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