{"paper":{"title":"The Brunn-Minkowski inequality and a Minkowski problem for $\\mathcal{A}$-harmonic Green's function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrew Vogel, John Lewis, Murat Akman, Olli Saari","submitted_at":"2018-10-08T23:58:42Z","abstract_excerpt":"In this article we study two classical problems in convex geometry associated to $\\mathcal{A}$-harmonic PDEs, quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace equation. Let $p$ be fixed with $2\\leq n\\leq p<\\infty$. For a convex compact set $E$ in $\\mathbb{R}^{n}$, we define and then prove the existence and uniqueness of the so called $\\mathcal{A}$-harmonic Green's function for the complement of $E$ with pole at infinity. We then define a quantity $\\mbox{C}_{\\mathcal{A}}(E)$ which can be seen as the behavior of this function near infinity.\n  In the first part of this art"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.03752","kind":"arxiv","version":1},"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"}