{"paper":{"title":"On the reconstruction of planar lattice-convex sets from the covariogram","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Barbara Langfeld, Gennadiy Averkov","submitted_at":"2010-11-24T23:15:49Z","abstract_excerpt":"A finite subset $K$ of $\\mathbb{Z}^d$ is said to be lattice-convex if $K$ is the intersection of $\\mathbb{Z}^d$ with a convex set. The covariogram $g_K$ of $K\\subseteq \\mathbb{Z}^d$ is the function associating to each $u \\in \\integer^d$ the cardinality of $K\\cap (K+u)$. Daurat, G\\'erard, and Nivat and independently Gardner, Gronchi, and Zong raised the problem on the reconstruction of lattice-convex sets $K$ from $g_K$. We provide a partial positive answer to this problem by showing that for $d=2$ and under mild extra assumptions, $g_K$ determines $K$ up to translations and reflections. As a c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.5530","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"}