A sharp Poincaré inequality for functions in 𝐖^{1,∞}(Ω;ℝ)

For each natural number n n and any bounded, convex domain Ω ⊂ R n \Omega \subset \mathbb {R}^n we characterize the sharp constant C ( n , Ω ) C(n,\Omega ) in the Poincaré inequality ‖ f − f ¯ Ω ‖ L ∞ ( Ω ; R ) ≤ C ( n , Ω ) ‖ ∇ f ‖ L ∞ ( Ω ; R ) \| f - \bar {f}_{\Omega }\|_{L^{\infty }(\Omega ;\mathbb {R})} \leq C(n,\Omega ) \|\nabla f\|_{L^{\infty }(\Omega ;\mathbb {R})} . Here, f ¯ Ω \bar {f}_{\Omega } denotes the mean value of f f over Ω \Omega . In the case that Ω \Omega is a ball B r B_r of radius r r in R n \mathbb {R}^n , we calculate C ( n , B r ) = C ( n ) r C(n,B_r)=C(n)r explicitly in terms of n n and a ratio of the volumes of the unit balls in R 2 n − 1 \mathbb {R}^{2n-1} and R n \mathbb {R}^n . More generally, we prove that C ( n , B r ( Ω ) ) ≤ C ( n , Ω ) ≤ n n + 1 d i a m ( Ω ) C(n,B_{r(\Omega )}) \leq C(n,\Omega ) \leq \frac {n}{n+1}\mathrm {diam}(\Omega ) , where B r ( Ω ) B_{r(\Omega )} is a ball in R n \mathbb {R}^n with the same n − n- dimensional Lebesgue measure as Ω \Omega . 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.