We prove the sharp Poincaré inequality ||u − med Ω (u)||∞ ≤ C(Ω)||∇u||∞ valid for all u in W^{1,∞} (Ω, R) and with constant C(Ω) = max_{x∈Ω} {R_x : |B(x, R_x) ∩ Ω| = |Ω|/2}. Here, Ω ⊂ R^d is a bounded, convex domain, and the median value of u is med Ω (u) = inf{t ∈ R : |{u > t}| ≤ |Ω|/2}. We find that the graphs of extremal u are cones. We also prove the inequalities diam (Ω)/2 ≤ C(Ω) ≤ diam (Ω) in which the lower bound is sharp, while the upper bound is capable of improvement provided C(Ω) = R_x where x lies in ∂Ω. When Ω is a ball B(x, r) ⊂ R^d we calculate C(B(x, r)) and argue that C(B(x, r)) → √2 r as d → ∞.