{"paper":{"title":"The Minkowski problem, new constant curvature surfaces in R^3, and some applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Antonio Alarcon, Rabah Souam","submitted_at":"2012-04-20T17:51:15Z","abstract_excerpt":"Let $m\\in\\mathbb{N},$ $m\\geq 2,$ and let $\\{p_j\\}_{j=1}^m$ be a finite subset of $\\mathbb{S}^2$ such that $0\\in\\mathbb{R}^3$ lies in its positive convex hull. In this paper we make use of the classical Minkowski problem, to show the complete family of smooth convex bodies $K$ in $\\mathbb{R}^3$ whose boundary surface consists of an open surface $S$ with constant Gauss curvature (respectively, constant mean curvature) and $m$ planar compact discs $\\bar{D_1},...,\\bar{D_m},$ such that the Gauss map of $S$ is a homeomorphism onto $\\mathbb{S}^2-\\{p_j\\}_{j=1}^m$ and $D_j\\bot p_j,$ for all $j.$\n  We d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.4687","kind":"arxiv","version":3},"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"}