A Study of Certain Sharp Poincaré Constants as Set Functions of Their Domain

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 [Bevan et al. in Proc Am Math Soc 151:1071–1085, 2023 (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 ) = max i | 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.