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Capacities, surface area, and radial sums. (English) Zbl 1163.52001

Summary: A dual capacitary Brunn-Minkowski inequality is established for the \((n - 1)\)-capacity of radial sums of star bodies in \(\mathbb R^n\). This inequality is a counterpart to the capacitary Brunn-Minkowski inequality for the \(p\)-capacity of Minkowski sums of convex bodies in \(\mathbb R^n\), \(1\leqslant p<n\), proved by Borell, Colesanti, and Salani. When \(n\geqslant 3\), the dual capacitary Brunn-Minkowski inequality follows from an inequality of Bandle and Marcus, but here a new proof is given that provides an equality condition.
Note that when \(n=3\), the \((n - 1)\)-capacity is the classical electrostatic capacity.
A proof is also given of both the inequality and a (different) equality condition when \(n=2\). The latter case requires completely different techniques and an understanding of the behavior of surface area (perimeter) under the operation of radial sum. These results can be viewed as showing that in a sense \((n - 1)\)-capacity has the same status as volume in that it plays the role of its own dual set function in the Brunn-Minkowski and dual Brunn-Minkowski theories.

MSC:

52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
52A40 Inequalities and extremum problems involving convexity in convex geometry
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
35J70 Degenerate elliptic equations
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