{"paper":{"title":"High-level convexity for products of squared Euclidean distance functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.CA","authors_text":"Cornel Pintea, George C. \\c{T}urca\\c{s}, Tudor Micu","submitted_at":"2026-05-29T19:42:40Z","abstract_excerpt":"We study smooth functions on Euclidean space whose Hessian is positive definite outside a bounded set, with emphasis on products of squared distance functions. More precisely, we first prove a simple convexity principle: if the superlevel region $f^{-1}([c,\\infty))$ is contained in the Hessian-positive region of $f$, then the sublevel set $\\{f\\le c\\}$ is convex. We apply this to finite products $F_P(x)=\\prod_{p\\in P}\\|x-p\\|^2$, proving that their Hessian-positive complements are bounded. For the two-centre product $F_{p,q}(x)=\\|x-p\\|^2\\|x-q\\|^2$ in dimension $n\\ge2$, we compute the Hessian-pos"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.00316","kind":"arxiv","version":1},"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/2606.00316/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"}