A sharp Poincare inequality for functions from $W^{1,\infty}$

For each natural number n and any bounded, convex domain $\Omega\subset\mathbb R^n$, we characterize the sharp constant in the Poincaré inequality for $L^\infty$-norms of a function and its first derivatives. We calculate explicitly the constant for the case of a unit ball and show that, among convex domains of equal measure, balls have the best, i.e. smallest, Poincaré constant.