{"paper":{"title":"The sum of squared logarithms inequality in arbitrary dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Christian Thiel, Lev Borisov, Patrizio Neff, Suvrit Sra","submitted_at":"2015-08-17T14:19:03Z","abstract_excerpt":"We prove the \\emph{sum of squared logarithms inequality} (SSLI) which states that for nonnegative vectors $x, y \\in \\mathbb{R}^n$ whose elementary symmetric polynomials satisfy $e_k(x)\\le e_k(y)$ (for $1\\le k < n$) and $e_n(x)=e_n(y)$, the inequality $\\sum_i (\\log x_i)^2 \\le \\sum_i (\\log y_i)^2$ holds. Our proof of this inequality follows by a suitable extension to the complex plane. In particular, we show that the function $f\\colon M\\subseteq \\mathbb{C}^n\\to \\mathbb{R}$ with $f(z)=\\sum_i(\\log z_i)^2$ has nonnegative partial derivatives with respect to the elementary symmetric polynomials of $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.04039","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"}