Memoirs of the American Mathematical Society

In this article we study two classical potential-theoretic problems in convex geometry. The first problem is an inequality of Brunn-Minkowski type for a nonlinear capacity, Cap A , \operatorname {Cap}_{\mathcal {A}}, where A \mathcal {A} -capacity is associated with a nonlinear elliptic PDE whose structure is modeled on the p p -Laplace equation and whose solutions in an open set are called A \mathcal {A} -harmonic. In the first part of this article, we prove the Brunn-Minkowski inequality for this capacity: \[ [ Cap A ⁡ ( λ E 1 + ( 1 − λ ) E 2 ) ] 1 ( n − p ) ≥ λ [ Cap A ⁡ ( E 1 ) ] 1 ( n − p ) + ( 1 − λ ) [