{"paper":{"title":"Inverse Additive Problems for Minkowski Sumsets II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"D. J. Grynkiewicz, G. A. Freiman, O. Serra, Y. Stanchescu","submitted_at":"2010-12-16T14:13:09Z","abstract_excerpt":"The Brunn-Minkowski Theorem asserts that $\\mu_d(A+B)^{1/d}\\geq \\mu_d(A)^{1/d}+\\mu_d(B)^{1/d}$ for convex bodies $A,\\,B\\subseteq \\R^d$, where $\\mu_d$ denotes the $d$-dimensional Lebesgue measure. It is well-known that equality holds if and only if $A$ and $B$ are homothetic, but few characterizations of equality in other related bounds are known. Let $H$ be a hyperplane. Bonnesen later strengthened this bound by showing $$\\mu_d(A+B)\\geq (M^{1/(d-1)}+N^{1/(d-1)})^{d-1}(\\frac{\\mu_d(A)}{M}+\\frac{\\mu_d(B)}{N}),$$ where $M=\\sup\\{\\mu_{d-1}((\\mathbf x+H)\\cap A)\\mid \\mathbf x\\in \\R^d\\}$ and $N=\\sup\\{\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.3610","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"}