{"paper":{"title":"The Brunn-Minkowski inequality and a Minkowski problem for nonlinear capacity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrew Vogel, Jasun Gong, Jay Hineman, John Lewis, Murat Akman","submitted_at":"2017-09-01T19:02:29Z","abstract_excerpt":"In this article we study two classical potential-theoretic problems in convex geometry corresponding to a nonlinear capacity, $\\mbox{Cap}_{\\mathcal{A}}$, where $\\mathcal{A}$-capacity is associated with a nonlinear elliptic PDE whose structure is modeled on the $p$-Laplace equation and whose solutions in an open set are called $ \\mathcal{A}$-harmonic.\n  In the first part of this article, we prove the Brunn-Minkowski inequality for this capacity: \\[ \\left[\\mbox{Cap}_\\mathcal{A}(\\lambda E_1 +(1-\\lambda)E_2)\\right]^{\\frac{1}{(n-p)}}\\geq\\lambda\\left[\\mbox{Cap}_\\mathcal{A}(E_1)\\right]^{\\frac{1}{(n-p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.00447","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}