A Note on the Poincaré Inequality for Convex Domains

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Abstract
A proof for the Poincaré inequality with explicit constant for convex domains is given. This proof is a modification of the original proof of L. F. Payne and H. F. Weinberger [Arch. Rat. Mech. Anal. 5 (1960) 286–292], which is valid only for the two-dimensional case.