The Brunn - Minkowski inequality and a Minkowski problem for -harmonic Green's function

In this article we study two classical problems in convex geometry associated to -harmonic PDEs, quasi-linear elliptic PDEs whose structure is modelled on the p-Laplace equation. Let p be fixed with 2 ≤ n ≤ p <. For a convex compact set E in n, we define and then prove the existence and uniqueness of the so-called -harmonic Green's function for the complement of E with pole at infinity. We then define a quantity C which can be seen as the behavior of this function near infinity. In the first part of this article, we prove that C satisfies the following Brunn-Minkowski-type inequality: [ C (λ E 1 + (1 - λ) E 2) ] 1 p - n ≥ λ [ C (E 1) ] 1 p - n + (1 - λ) [ C (E 2) ] 1 p - n when n < p < 0 ≤ λ ≤ 1, and E 1, E 2 are nonempty convex compact sets in n. While p = n {p=n} then C (λ E 1 + (1 - λ) E 2) ≥ λ C (E 1) + (1 - λ) C (E 2), where 0 ≤ λ ≤ 1 and E 1, E 2 are convex compact sets in n containing at least two points. Moreover, if equality holds in the either of the above inequalities for some E 1 and E 2, then under certain regularity and structural assumptions on we show that these two sets are homothetic. The classical Minkowski problem asks for necessary and sufficient conditions on a non-negative Borel measure on the unit sphere n - 1 to be the surface area measure of a convex compact set in n under the Gauss mapping for the boundary of this convex set. In the second part of this article we study a Minkowski-type problem for a measure associated to the -harmonic Green's function for the complement of a convex compact set E when n ≤ p <. If μ E denotes this measure, then we show that necessary and sufficient conditions for existence under this setting are exactly the same conditions as in the classical Minkowski problem. Using the Brunn-Minkowski inequality result from the first part, we also show that this problem has a unique solution up to translation.