BRUNN-MINKOWSKI INEQUALITY FOR p-HARMONIC MEASURES: Research Paper Podcast

Alex: This blended height from supremal convolution lower-bounds the real p-harmonic on the mix. And that implies something about gradients or the measure?

Sam: Exactly—the true function exceeds the convolution inside, so its gradients are steeper near the boundary. Support functions add linearly under Minkowski sums, so sublevel sets—regions where the function exceeds height t—inherit additivity. The convolution keeps enough convexity in sublevels to make boundary gradients grow more than additively, boosting the p-harmonic measure and making T larger than the weighted average.

Alex: The inequality for T follows from that gradient growth.

Sam: Yes, and they link T(K) to a boundary limit: as you approach the vanishing boundary from inside, T equals the limit of a surface integral over sublevel boundaries of (u divided by distance to that boundary)^{p-1} times area. Boundary Harnack principles and Green's representations justify this, tying the measure to near-boundary behavior.

Alex: That grounds T in the function's shape near the edge.

Alex: With this setup, how do they define the local version around a fixed reference shape?

Sam: They pick a compact convex set K0 with smooth boundary in a neighborhood N, and a reference p-harmonic u0—positive inside K0 intersect N, zero on K0's boundary. For nearby K in a local family, like K0 plus a scaled bit of another set, they solve uK: it satisfies the p-Laplacian inside K intersect N, vanishes on ∂K inside N, and matches u0 on the outer edge. The measures μK come from |∇uK|^{p-1} on ∂K, pushed to the sphere by the Gauss map. T(K) integrates the support function hK over μK, with homogeneity under scaling.

Alex: That stages the inequality on local sums. The proof hinges on the supremal convolution being a p-subsolution on the blend?

Sam: Yes. They rewrite the p-Laplacian using support functions of sublevel sets {u > t}—convex bodies shrinking as t rises, like layered slices of a cake. The support function h_u(y, t) measures max projection of a layer in direction y. Key links: at x, h_u along the inward normal at height u(x) recovers position dot normal; its gradient is x; t-derivative is minus 1 over |∇u|.

Alex: These h_u capture all layers' geometry. And support functions add linearly on Minkowski sums of sublevels.

Sam: Precisely. Supremal convolution's sublevels mix via inf over decompositions then sup—like picking the tallest short glass from pairs averaging to a spot—preserving convexity so h for the blend fits the subsolution form for the p-Laplacian. Comparison lifts the true u above it, steepening gradients and superadding the measure to prove T's power concave locally.

Alex: How does that formula show the blended function v acts as a subsolution?

Sam: For subsolution, the p-Laplacian of v must be ≤0—its flow bends more sharply. They use relations: gradient of h_u in inward normal gives position x; Hessian relates to inverse of shape operator W_u, measuring boundary bend like surface curvature. The p-Laplacian becomes trace of that inverse plus h_u times identity, scaled by powers of ∂_t h_u = -1/|∇u|, plus other terms.

Alex: It's averaging curvatures from layer supports, with factors from height changes. For v, suprema over minima make this expression non-positive?

Sam: Yes. Supremal convolution makes h_v exceed weighted inf-convolution of originals, preserving matrix convexity. Hessians add under true sums, but sup-min ensures inverse trace grows superadditively while derivatives align, so Δ_p v ≤0 pointwise. Comparison steepens true gradients. Boundary Harnack—|∇u| between constants times u/distance—lets them confirm T's limit via co-area integrals and Green's representations.

Alex: That ties layer curvatures to stronger measures on blends.

Alex: How do they nail the boundary limit for T, handling edge singularities?

Sam: They use Green's functions—like steady heat flow tools with fixed edges. Near a sublevel boundary point, a test function φ_s pulls residues from singularities via principal value integrals and sphere averages. Dominated convergence takes s-to-zero limit, yielding T as surface measure; co-area slices volumes to layers, integration by parts swaps to boundaries.

Alex: Green's reps extract boundary behavior. p>2 ensures estimates near the edge?

Sam: Correct. For subsolutions with fixed u0, they shrink N so sublevels stay convex. Supremal convolution on blend K_λ has sublevels as exact Minkowski blends; layer supports average linearly. Matrices from second derivatives plus identity times height, vectors from gradients, scalars from height change—a lemma shows convex blends make traces and forms superadditive, so plugged-in expression ≤0, proving subsolution. Comparison steepens gradients.

Alex: Homogeneity follows by rescaling.

Alex: The proof shows T on the blend beats the minimum of originals via distance estimates on sublevels—projections to edges and curvature comparisons. Homogeneity scales to powered concavity?

Sam: Yes—dilate sets to match T at reference, blend and rescale back for the inequality.

Alex: Worth noting open questions or limits?

Sam: It holds only near smooth K0—global versions elusive due to vanishing boundary. Equality cases and uniqueness not fully characterized; p-harmonic measures aren't always unique. Still, it opens paths to uniqueness in p-harmonic Minkowski problems and broader cases.

Alex: A meaningful local bridge with clear next steps. It shows how these nonlinear measures echo classical geometry nearby. Thanks for joining us on ResearchPod.