2021/11/04 by Takashi Satomi, Satomi, Takashi · 1 citation
Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Group Theory (math.GR) #Limits and Structures in Graph Theory #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2111.15349
openalex publication_date 2021/11/04 · openalex created_date 2021/12/06 · openalex updated_date 2026/07/28
Let m(G) be the infimum of the volumes of all open subgroups of a unimodular locally compact group G. Suppose integrable functions ϕ1 , ϕ2 \colon G → [0,1] satisfy ‖ ϕ1 ‖ ≤ ‖ ϕ2 ‖ and ‖ ϕ1 ‖ + ‖ ϕ2 ‖ ≤ m (G), where ‖ ⋅ ‖ denotes the L1-norm with respect to a Haar measure dg on G. We have the following inequality for any convex function f \colon [0, ‖ ϕ1 ‖ ] → ℝ with f(0) = 0: ∫G f ∘ ( ϕ1 * ϕ2 ) (g) dg ≤ 2 ∫0‖ ϕ1 ‖ f(y) dy + ( ‖ ϕ2 ‖ - ‖ ϕ1 ‖ ) f( ‖ ϕ1 ‖ ). As a corollary, we have a slightly stronger version of Brunn-Minkowski-Kemperman inequality. That is, we have vol_* ( B1 B2 ) ≥ vol ( \ g ∈ G | 1B1 * 1B2 (g) gt; 0 \ ) ≥ vol (B1) + vol (B2) for any non-null measurable sets B1 , B2 ⊂ G with vol (B1) + vol (B2) ≤ m(G), where vol_* denotes the inner measure and 1B the characteristic function of B.