A Lp Brunn-Minkowski Theory for Logarithmic Capacity

Abstract

In the present paper, we first introduce the concepts of the Lp n-dimensional Logarithmic-Capacity measure and Lp mixed n-dimensional Logarithmic-Capacity and then prove some geometric properties of Lp n-dimensional Logarithmic-Capacity measure and a Lp Minkowksi inequality for the n-dimensional Logarithmic-Capacity for any fixed n ≥ 2 and p ≥ 1. As an application, we establish a Hadamard variational formula for the n-dimensional Logarithmic-Capacity under p-sum for any fixed n ≥ 2 and p ≥ 1, which extends some results of Borell (Ann. Sci. Ecole Norm. Sup. 17(3), 451–467 1984), Jerison (Adv. Math. 122(2), 262–279 1996), Colesanti-Cuoghi (Potential Anal. 22(3), 289–304 2005), Akman-Lewis-Saari-Vogel (in press) and Xiao (2013, 2018). With the Hadamard variational formula and Minkowksi inequality mentioned above, we prove the existence and uniqueness of the solution for the Lp Minkowski problem for the Logarithmic-Capacity.