{"paper":{"title":"A Brunn-Minkowski theory for coconvex sets of finite volume","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Rolf Schneider","submitted_at":"2017-06-12T11:34:48Z","abstract_excerpt":"Let $C$ be a closed convex cone in ${\\mathbb R}^n$, pointed and with interior points. We consider sets of the form $A=C\\setminus A^\\bullet$, where $A^\\bullet\\subset C$ is a closed convex set. If $A$ has finite volume (Lebesgue measure), then $A$ is called a $C$-coconvex set. The family of $C$-coconvex sets is closed under the addition $\\oplus$ defined by $C\\setminus(A_1\\oplus A_2)= (C\\setminus A_1)+(C\\setminus A_2)$. We develop first steps of a Brunn--Minkowski theory for $C$-coconvex sets, which relates this addition to the notion of volume. In particular, we establish the equality conditions"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.03573","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"}