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

For bounded, convex sets $$\Omega \subset \mathbb {R}^d$$ Ω ⊂ R d , the sharp Poincaré constant $$C(\Omega )$$ C ( Ω ) , which appears in $$||f-\bar{f}_{_{\Omega }}||_{L^{\infty }(\Omega )} \le C(\Omega )||\nabla f||_{L^{\infty }(\Omega )}$$ | | f - f ¯ Ω | | L ∞ ( Ω ) ≤ C ( Ω ) | | ∇ f | | L ∞ ( Ω ) , is given by $$C(\Omega )=\max _{_{\partial \Omega }}\zeta $$ C ( Ω ) = max ∂ Ω ζ for a specific convex function $$\zeta $$ ζ [Bevan et al. in Proc Am Math Soc 151:1071–1085, 2023 (Theorem 1.1)]. We study $$C(\cdot )$$ C ( · ) as a function on convex sets, in particular on polyhedra, and find that while a geometric characterization of $$C(\Omega )$$ C ( Ω ) for triangles is possible, for other polyhedra the problem of ordering $$\zeta (V_i)$$ ζ ( V i ) , where $$V_i$$ V i are the vertices of $$\Omega $$ Ω , can be formidable. In these cases, we develop estimates of $$C(\Omega )$$ 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 $$V_i$$ V i and centroid $$\gamma (Q)$$ γ ( Q ) is the quantity $$D(Q)=\max _{i}|V_i-\gamma (Q)|$$ D ( Q ) = max i | V i - γ ( Q ) | , with an error of up to $$\sim 8\%$$ ∼ 8 % . A numerical study suggests that a similar statement holds for k-gons, this time with a maximal error across all k-gons of $$\sim 13\%$$ ∼ 13 % . We explore the question of whether there is, for each $$\Omega $$ Ω , at least one point M capable of ordering the $$\zeta (V_i)$$ ζ ( V i ) according to the ordering of the $$|V_i-M|$$ | V i - M | . For triangles, M always exists; for quadrilaterals, M seems always to exist; for 5-gons and beyond, they seem not to.