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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.04418","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2026-07-05T17:19:59Z","cross_cats_sorted":[],"title_canon_sha256":"11fa2532d8d355e7f1bf20aa75d40d9623f9d811f01681158e4bd356d2711343","abstract_canon_sha256":"a28fad0b7744e941006fafc5fcf6a05dec575a2643cbeb65eaaf17a0a28e9e23"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:19:17.545027Z","signature_b64":"le1PzwJFdP9fABXoTC+PN3ShMuECS3IjoSHY88ZG6rF6QWdhK9UNZOU6/JVEWI5b7pOUcIIbdSnfW68qK3lUDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e6204ade589c16ed4ceeb8cce2a49ccd956c95c824d4692f0cb141b2a4a398dd","last_reissued_at":"2026-07-07T02:19:17.544288Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:19:17.544288Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $K$. 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