{"paper":{"title":"Impossibility of a nontrivial Brunn--Minkowski inequality for higher Dirichlet eigenvalue","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Khai-Hoan Nguyen-Dang, Tr\\'i Minh L\\^e","submitted_at":"2026-07-05T17:19:59Z","abstract_excerpt":"Let $\\lambda_j(K)$ be the $j$th Dirichlet eigenvalue of a convex body $K$. It is well known that $\\lambda_1$ satisfies a Brunn--Minkowski inequality: $K \\mapsto \\lambda_1(K)^{-1/2}$ is concave on the family of convex bodies. We show that no analogous statement holds for higher eigenvalues. More precisely, for any $j \\geq 2$ and $N \\geq 2$, if $K \\mapsto (f \\circ \\lambda_j)(K)$ is concave on the family of convex bodies in $\\mathbb{R}^N$ for some function $f: (0, \\infty) \\to \\mathbb{R}$, then $f$ must be constant."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04418","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/2607.04418/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"}