Bonnesen-type inequalities for surfaces of constant curvature

Abstract

A Bonnesen-type inequality is a sharp isoperimetric inequality that includes an error estimate in terms of inscribed and circumscribed regions. A kinematic technique is used to prove a Bonnesen-type inequality for the Euclidean sphere (having constant Gauss curvature κ > 0) and the hyperbolic plane (having constant Gauss curvature κ < 0). These generalized inequalities each converge to the classical Bonnesen-type inequality for the Euclidean plane as κ → 0.

An isoperimetric deficit upper bound of the convex domain in a surface of constant curvature

“…The lower bounds of the isoperimetric deficit are also called the Bonnesen-style inequalities. Klain, Zhou and Chen obtain the generalized isoperimetric inequalities and some Bonnesen-style inequalities for domains in a two-dimensional surface X of constant curvature (see [7,19]). …”

Isometry-Invariant Valuations on Hyperbolic Space

“…120]), a generalization of the classical isoperimetric inequality (see also [Os]). Variations of these kinematic techniques can also be found in [Kl3] and [San,p. 324].…”

“…It would be interesting to see how a suitable variation of Corollary 4.4 (possibly using integration over a suitable chosen proper subgroup of isometries) might yield hyperbolic analogues of Minkowski's mixed volumes and the related Brunn-Minkowski theory [Sc1]. Corollary 4.4 and its higher-dimensional generalizations have numerous applications to questions in geometric probability, leading, for example, to hyperbolic analogues of Hadwiger's containment theorem for planar regions [KR], [San] and to Bonnesen's inequality for area (see [Kl3] and [San,p. 120]), a generalization of the classical isoperimetric inequality (see also [Os]).…”

“…Such characterizations have had fundamental impact in convex, integral, and combinatorial geometry [Al1]- [Al3], [Ha], [KR], [Kl2], [Kl3], [Lud], [LR], [Sc2], [McM1]- [McM3] as well as to the theory of dissection of polytopes [Bo], [Ha], [KR], [McM3], [Sah].…”

ON BONNESEN-STYLE ALEKSANDROV-FENCHEL INEQUALITIES IN ℝn

“…This was done much later, first by Hadwiger [21] for n = 3, and then by Dinghas [7] for arbitrary dimension. Although it is a hard work to obtain some Bonnesenstyle inequalities in higher dimensional space, mathematicians are still working on finding unknown invariants of geometric significance (see [9,10,18,20,22,25,31,32,35,38,39,40,41,42,43,44,45,46,47,48]). …”