Abstract
For each natural number n and any bounded, convex domain Ω ⊂ R n we characterize the sharp constant C(n, Ω) in the Poincaré inequality ||f − ¯ f Ω || L ∞ (Ω;R) ≤ C(n, Ω)|||f || L ∞ (Ω;R). Here, ¯ f Ω denotes the mean value of f over Ω. In the case that Ω is a ball Br of radius r in R n , we calculate C(n, Br) = C(n)r explicitly in terms of n and a ratio of the volumes of the unit balls in R 2n−1 and R n. More generally, we prove that C(n, B r(Ω)) ≤ C(n, Ω) ≤ n n+1 diam(Ω), where B r(Ω) is a ball in R n with the same n−dimensional Lebesgue measure as Ω. Both bounds are sharp, and the lower bound can be interpreted as saying that, among convex domains of equal measure, balls have the best, i.e. smallest, Poincaré constant.