{"paper":{"title":"$p$-Means of Convex Bodies: Sharpening Relations and Structural Properties","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Florian Grundbacher, Ren\\'e Brandenberg","submitted_at":"2023-12-29T08:19:00Z","abstract_excerpt":"We study general $p$-means of convex bodies, extending the classical definitions by W. J. Firey via support and gauge functions to two families ranging over all $p \\in [-\\infty,\\infty]$. For values of $p$ beyond the classical ranges, we show that $p$-means of polytopes are again polytopes, yielding simpler structural descriptions. Using a natural characterization of dilates of convex bodies based on their boundary structure, we characterize the equality cases between the two types of $p$-means for the same $p$-value. Extending recent results on standard mean-symmetrizations of convex bodies, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2312.17512","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2312.17512/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}