Sharp lower bounds for Poincaré-Sobolev constants λ_{p,q}(Ω) on convex Ω are given explicitly in terms of perimeter and volume powers, with new optimal Lebesgue-norm bounds on the distance function from the boundary.
van den Berg, Estimates for the torsion function and Sobolev constants,Potential Anal.36(2012), 607–616
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Sharp Makai-type inequalities for the best Poincar\'e-Sobolev constants
Sharp lower bounds for Poincaré-Sobolev constants λ_{p,q}(Ω) on convex Ω are given explicitly in terms of perimeter and volume powers, with new optimal Lebesgue-norm bounds on the distance function from the boundary.