For bounded, convex sets Ω ⊂ R d , the sharp Poincaré constant C(Ω), which appears in ||f − f_Ω || L^∞(Ω) ≤ C(Ω)|||f || L^∞(Ω) , is given by C(Ω) = max ∂Ω ζ for a specific convex function ζ ([3, Theorem 1.1]). We study C(·) as a function on convex sets, in particular on polyhedra, and find that while a geometric characterization of C(Ω) for triangles is possible, for other polyhedra the problem of ordering ζ(V_i), where V_i are the vertices of Ω, can be formidable. In these cases, we develop estimates of C(Ω) from above and below in terms of more tractable quantities. We find, for example, that a good proxy for C(Q) when Q is a planar polygon with vertices V_i and centroid γ(Q) is the quantity D(Q) = maxi |V_i − γ(Q)|, with an error of up to ∼ 8%. A numerical study suggests that a similar statement holds for k−gons, this time with a maximal error across all k−gons of ∼ 13%. We explore the question of whether there is, for each Ω, at least one point M capable of ordering the ζ(V_i) according to the ordering of the |V_i − M |. For triangles, M always exists; for quadrilaterals, M seems always to exist; for 5−gons and beyond, they seem not to.