Valuations and Surface Area Measures

By Christoph Haberl, Lukas Parapatits. 2014

69 citations recorded by the source corpus.

We consider valuations defined on polytopes containing the origin which have measures on the sphere as values. We show that the classical surface area measure is essentially the only such valuation which is SL(n) contravariant of degree one. Moreover, for all real $p$, an $L_p$ version of the above result is established for GL(n) contravariant valuations of degree $p$. This provides a characterization of the $L_p$ surface area measures from the $L_p$ Brunn-Minkowski theory.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c613a2bbdd88600e3482bd0d71963ea921412bc7

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In Applied Mathematics

1. Yiming Zhao
2. Deping Ye
3. Yong Huang
4. K. Boroczky
5. Dongmeng Xi
6. Deane Yang
7. Gaoyong Zhang
8. Erwin Lutwak

In Geometry and Topology

1. Yiming Zhao
2. Yong Huang
3. K. Boroczky
4. Deane Yang
5. Gaoyong Zhang
6. Erwin Lutwak
7. Dongmeng Xi
8. Deping Ye

Citations in the corpus

• The dual Minkowski problem for symmetric convex bodies (2017, background): “bodies) that are dual to their counterparts in the Brunn-Minkowski theory. The latter have been studied extensively in recent years. See Haberl [22], Haberl & Ludwig [23], Haberl & Parapatits [24,25], Ludwig [34,35], Ludwig & Reitzner [36], Schuster [45,46], Schuster & Wannerer [47],”
• THE ORLICZ-BRUNN-MINKOWSKI THEORY: A GENERAL FRAMEWORK, ADDITIONS, AND INEQUALITIES (2013, background): “he remainder of this introduction. In particular, in the Lp-Brunn-Minkowski theory, S(K,·) is replaced by the p-surface area measure Sp(K,·) given by dSp(K,·) = h 1−p K dS(K,·). Haberl and Parapatits [12] provide a characterization of these measures. In Section 8 we introduce a new notion of Orlicz ”
• The Minkowski problem in Gaussian probability space (2020, background): “In fact, Haberl-Parapatits [32] gave a valuation characterization of surface area measure.”
• The dual Minkowski problem for negative indices (2017, background): “See, e.g., Haberl [19], Haberl & Ludwig [20] ,Haberl & Parapatits [21,22], Ludwig [27,28], Ludwig & Reitzner [29], Schuster [39, 40], Schuster & Wannerer [41] and the references therein for important valuations and their characterizations in the theory of convex bodies.”