The Brunn-Minkowski Inequality and A Minkowski Problem for Nonlinear Capacity

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: \[ [ Cap A ⁡ ( λ E 1 + ( 1 − λ ) E 2 ) ] 1 ( n − p ) ≥ λ [ Cap A ⁡ ( E 1 ) ] 1 ( n − p ) + ( 1 − λ ) [ Cap A ⁡ ( E 2 ) ] 1 ( n − p ) \left [\operatorname {Cap}_\mathcal {A} ( \lambda E_1 + (1-\lambda ) E_2 )\right ]^{\frac {1}{(n-p)}} \geq \lambda \, \left [\operatorname {Cap}_\mathcal {A} ( E_1 )\right ]^{\frac {1}{(n-p)}} + (1-\lambda ) \left [\operatorname {Cap}_\mathcal {A} (E_2 )\right ]^{\frac {1}{(n-p)}} \] when 1 > p > n , 0 > λ > 1 , 1>p>n, 0 > \lambda > 1, and E 1 , E 2 E_1, E_2 are convex compact sets with positive A \mathcal {A} -capacity. Moreover, if equality holds in the above inequality for some E 1 E_1 and E 2 , E_2, then under certain regularity and structural assumptions on