On the isodiametric and isominwidth inequalities for planar bisections

Abstract

For a given planar convex compact set K, consider a bisection {A, B} of K (i.e., A ∪ B = K and whose common boundary A ∩ B is an injective continuous curve connecting two boundary points of K) minimizing the corresponding maximum diameter (or maximum width) of the regions among all such bisections of K.In this note we study some properties of these minimizing bisections and we provide analogous to the isodiametric (Bieberbach, 1915), the isominwidth (Pál, 1921), the reverse isodiametric (Behrend, 1937), and the reverse isominwidth (González Merino & Schymura, 2018) inequalities.Surprisingly enough, the optimal set in the general situation, described in Theorem 2, is still the same set as in Proposition 1. This fact strengthens the idea that central symmetry is an inherent

On Blaschke–Santaló diagrams for the torsional rigidity and the first Dirichlet eigenvalue

“…The exponent 1 on η, α E , and A E in the right-hand sides of (1.2), (1.4), and (1.6) is sharp, in the sense that these quantitative inequalities are no longer true if the exponent is lowered. Moreover, the constants (though not optimal) are explicit and can be taken equal to the following ones: 5), and…”

“…As far as we are aware, these are the only known proofs. This Pál inequality is sometimes also referred to as isominwidth inequality [5] due to its striking resemblance to a more ancient and studied inequality: the isoperimetric inequality (see for instance [8] and references therein). It's worth noting that replacing the perimeter constraint with the minimal width introduces several significant differences.…”