{"paper":{"title":"Compact convex sets of the plane and probability theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.PR","authors_text":"David Renault, Jean-Fran\\c{c}ois Marckert","submitted_at":"2012-08-27T14:33:42Z","abstract_excerpt":"The Gauss-Minkowski correspondence in $\\mathbb{R}^2$ states the existence of a homeomorphism between the probability measures $\\mu$ on $[0,2\\pi]$ such that $\\int_0^{2\\pi} e^{ix}d\\mu(x)=0$ and the compact convex sets (CCS) of the plane with perimeter~1. In this article, we bring out explicit formulas relating the border of a CCS to its probability measure. As a consequence, we show that some natural operations on CCS -- for example, the Minkowski sum -- have natural translations in terms of probability measure operations, and reciprocally, the convolution of measures translates into a new notio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.5408","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"}