Renormalization of potentials and generalized centers

Abstract

We generalize the Riesz potential of a compact domain in ${\mathbb{R}}^m$ by introducing a renormalization of the $r^{\alpha -m}$-potential for $\alpha \le 0$. This can be considered as generalization of the dual mixed volumes of convex bodies as introduced by Lutwak. We then study the points where the extreme values of the (renormalized) potentials are attained. These points can be considered as a generalization of the center of mass. We also show that only balls give extreme values among bodied with the same volume.

Adv. Appl. Math. 48 (2012), 365--392 Figure 11 has been corrected after publication. Theorem 3.12 and the exposition of Lemma 2.15 are modified in version 8

References in corpus (3)

• Extremal problems for the solid angle
• Möbius invariant energies and average linking with circles
• Extremal problems for the central projection

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• Geometric estimation of a potential and cone conditions of a body
• The incenter of a triangle as a cone isoperimetric center
• Stationary radial centers and characterization of convex polyhedrons
• Experimental investigation on the uniqueness of a center of a body
• Spherical Geometry and the Least Symmetric Triangle