Memoirs of the American Mathematical Society
Akman, Murat, Gong, Jasun, Hineman, Jay, Lewis, John, Vogel, Andrew · 2026 · 662 citationsRead the paper
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 ) [
What the full text shows
Theory or simulation- Proven Brunn-Minkowski inequality for nonlinear A-capacity. — [CapA(λE1 + (1 − λ)E2)]^(1/(n−p)) ≥ λ[CapA(E1)]^(1/(n−p)) + (1 − λ)[CapA(E2)]^(1/(n−p)) (1 < p < n, 0 < λ < 1, E1 and E2 are convex compact sets with positive A-capacity)
- Existence and uniqueness of a compact convex set E for the Minkowski problem associated with A-capacity. — Unique up to translation if p ≠ n − 1; unique up to translation and dilation if p = n − 1 (μ is a positive finite Borel measure on Sn−1 satisfying ∫ Sn−1 |〈θ, ζ〉| dμ(ζ) > 0 and ∫ Sn−1 ζ dμ(ζ) = 0)
- Setup
- Theoretical analysis in Euclidean n-space (Rn) for compact convex sets and Borel measures on the unit sphere (Sn−1).
- How
- The authors use potential-theoretic methods and nonlinear elliptic PDEs (A-harmonic functions) to prove a Brunn-Minkowski type inequality and solve the corresponding Minkowski problem via the Hadamard variational formula.
- Limitations
- Requires specific regularity and structural assumptions on A for the equality case of the Brunn-Minkowski inequality. · The Minkowski problem solution for p = n − 1 requires a constant b and a normalization of capacity (CapA(E) = 1) for uniqueness.
Summarized by Gemma 4 from the open-access full text · 2 Oct 2026. Read it