The Brunn-Minkowski inequality for \(p\)-capacity of convex bodies
Let \(K_1\), \(K_2\) be compact convex sets in \(\mathbb{R}^n\) with non-empty interior, and let \(\text{Vol}_n\) denote \(n\)-dimensional Lebesgue measure. If \(0\leq\lambda\leq 1\), then the Minkowski sum \(\lambda K_1+ (1-\lambda)K_2\) is defined by \(\{\lambda y+ (1- \lambda)z: y\in K_1, z\in K_2\}\). The original form of the Brunn-Minkowski inequality asserts that \([\text{Vol}_n(\cdot)]^{1/n}\) is concave with respect to Minkowski addition; that is, \([\text{Vol}_n(\lambda K_1 +(1- \lambda)K_2)]^{1/n}\) is at least as large as \(\lambda[\text{Vol}_n(K_1)]^{1/n}+ (1- \lambda) [\text{Vol}_n(K_2)]^{1/n}\). Appropriate analogues for Newtonian, and then logarithmic, capacity were subsequently established by \textit{C. Borell} [Math. Ann. 263, 179--184 (1984; Zbl 0546.31001); Ann. Sci. Éc. Norm. Supér., IV. Sér. 17, 451--467 (1984; Zbl 0573.60067)]. The present paper deals more generally with \(p\)-capacity \(\text{Cap}_p\), and shows that \([\text{Cap}_p(\cdot)]^{1/(n- p)}\) is concave with respect to Minkowski addition. (The case of Newtonian capacity corresponds to \(p= 2\).) Further, it is shown that equality holds if and only if \(K_2\) is a translate and dilate of \(K_1\). In the case of Newtonian capacity this last fact had previously been established by \textit{L. A. Caffarelli}, \textit{D. Jerison} and \textit{E. H. Lieb} [Adv. Math. 117, No. 2, 193--207 (1996; Zbl 0847.31005)].
scientific article; zbMATH DE number 2078339