{"paper":{"title":"A characterization of some mixed volumes via the Brunn-Minkowski inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Andrea Colesanti, Daniel Hug, Eugenia Saorin Gomez","submitted_at":"2012-02-10T12:03:04Z","abstract_excerpt":"We consider a functional $\\mathcal F$ on the space of convex bodies in $\\R^n$ defined as follows: ${\\mathcal F}(K)$ is the integral over the unit sphere of a fixed continuous functions $f$ with respect to the area measure of the convex body $K$. We prove that if $\\mathcal F$ satisfies an inequality of Brunn--Minkowski type, then $f$ is the support function of a convex body, i.e., $\\mathcal F$ is a mixed volume. As a consequence, we obtain a characterization of translation invariant, continuous valuations which are homogeneous of degree $n-1$ and satisfy a Brunn--Minkowski type inequality."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.2241","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"}