{"paper":{"title":"Covariograms generated by valuations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.MG","authors_text":"Gabriele Bianchi, Gennadiy Averkov","submitted_at":"2013-07-05T06:30:38Z","abstract_excerpt":"Let \\phi be a real-valued valuation on the family of compact convex subsets of \\mathbb{R}^n and let K be a convex body in \\mathbb{R}^n. We introduce the \\phi -covariogram g_{K,\\phi} of K as the function associating to each x \\in \\mathbb{R}^n the value \\phi(K \\cap (K+x)). If \\phi is the volume, then g_{K,\\phi} is the covariogram, extensively studied in various sources. When \\phi is a quermassintegral (e.g., surface area or mean width) g_{K,\\phi} has been introduced by Nagel.\n  We study various properties of \\phi -covariograms, mostly in the case n=2 and under the assumption that \\phi is transla"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.1529","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"}