2013/09/24 by Steinerberger, Stefan
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1309.6211
Let Ω⊂ ℝn be a convex. If u: Ω→ ℝ has mean 0, then we have the classical Poincaré inequality ‖u ‖Lp ≤ cp diam(Ω) ‖ ∇ u ‖Lp with sharp constants c2 = 1/π (Payne & Weinberger, 1960) and c1 = 1/2 (Acosta & Duran, 2005) independent of the dimension. The sharp constants cp for 1 < p < 2 have recently been found by Ferone, Nitsch & Trombetti (2012). The purpose of this short paper is to prove a much stronger inequality in the endpoint L1: we combine results of Cianchi and Kannan, Lovász & Simonovits to show that ‖u‖L1(Ω) ≤ \frac2log2 M(Ω) ‖∇ u‖L1(Ω) where M(Ω) is the average distance between a point in Ω and the center of gravity of Ω. If Ω is a simplex, this yields an improvement by a factor of ∼ √(n) in n dimensions. By interpolation, this implies that that for every convex Ω⊂ ℝn and every u:Ω→ ℝ with mean 0 ‖u‖Lp(Ω)≤ (\frac2log2 M(Ω) )(1)/(p)diam(Ω)1-(1)/(p)‖∇ u‖Lp(Ω).