{"paper":{"title":"Energy integrals and metric embedding theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Dami\\'an Pinasco, Daniel Carando, Daniel Galicer","submitted_at":"2013-12-03T01:17:54Z","abstract_excerpt":"For some centrally symmetric convex bodies $K\\subset \\mathbb R^n$, we study the energy integral $$ \\sup \\int_{K} \\int_{K} \\|x - y\\|_r^{p}\\, d\\mu(x) d\\mu(y), $$ where the supremum runs over all finite signed Borel measures $\\mu$ on $K$ of total mass one. In the case where $K = B_q^n$, the unit ball of $\\ell_q^n$ (for $1 < q \\leq 2$) or an ellipsoid, we obtain the exact value or the correct asymptotical behavior of the supremum of these integrals. We apply these results to a classical embedding problem in metric geometry. We consider in $\\mathbb R^n$ the Euclidean distance $d_2$. For $0 < \\alpha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.0678","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"}