{"paper":{"title":"On the tightness of Gaussian concentration for convex functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Petros Valettas","submitted_at":"2017-06-28T19:09:52Z","abstract_excerpt":"The concentration of measure phenomenon in Gauss' space states that every $L$-Lipschitz map $f$ on $\\mathbb R^n$ satisfies \\[ \\gamma_{n} \\left(\\{ x : | f(x) - M_{f} | \\geqslant t \\} \\right) \\leqslant 2 e^{\n  - \\frac{t^2}{ 2L^2} }, \\quad t>0, \\] where $\\gamma_{n} $ is the standard Gaussian measure on $\\mathbb R^{n}$ and $M_{f}$ is a median of $f$. In this work, we provide necessary and sufficient conditions for when this inequality can be reversed, up to universal constants, in the case when $f$ is additionally assumed to be convex. In particular, we show that if the variance ${\\rm Var}(f)$ (wi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.09446","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":""},"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"}