{"paper":{"title":"On the inner cone property for convex sets in two-step Carnot groups, with applications to monotone sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Daniele Morbidelli","submitted_at":"2018-08-20T15:24:10Z","abstract_excerpt":"In the setting of step two Carnot groups, we show a \"cone property\" for horizontally convex sets. Namely we prove that, given a horizontally convex set $C$, a pair of points $P\\in \\partial C$ and $Q\\in $ int $C$, both belonging to a horizontal line $\\ell$, then an open truncated subRiemannian cone around $\\ell$ and with vertex at $P$ is contained in $C$. We apply our result to the problem of classification of horizontally monotone sets in Carnot groups. We are able to show that monotone sets in the direct product $\\mathbb{H} \\times\\mathbb{R}$ of the Heisenberg group with the real line have hyp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.06513","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"}