Bonnesen-type inequalities for surfaces of constant curvature

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. Introduction A Bonnesen-type inequality is a sharp isoperimetric inequality that includes an error estimate in terms of inscribed and circumscribed regions. The classical example runs as follows: Suppose that K is a compact convex set in R. Denote by AK and PK the area and perimeter of K respectively. Let RK denote the circumradius of K, and let rK denote the inradius of K. Then P 2 K − 4πAK ≥ π(RK − rK). (1) The classical isoperimetric inequality immediately follows, namely, P 2 K − 4πAK ≥ 0, (2) with equality if and only if RK = rK , that is, if and only if K is a Euclidean disc. Proofs of these inequalities, along with variations and generalizations, can be found in any of [Bon24, San76, Oss79], for example. In this note the kinematic methods of Santaló and Hadwiger are used to prove Bonnesen-type inequalities for the Euclidean sphere (having constant Gauss curvature κ > 0) and the hyperbolic plane (having constant Gauss curvature κ < 0). Section 1 outlines necessary background material from integral geometry. In Section 2 we derive the first of the two main theorems in this article, a Bonnesen-type inequality for the sphere, stated in Theorem 2.1. The second main theorem of this article, Theorem 3.1, is a Bonnesen-type inequality for the hyperbolic plane, derived in Section 3. The limiting case as κ → 0 in either of Theorems 2.1 and 3.3 yields the classical Bonnesen inequality (1), as described above. A brief and direct proof of (1) using kinematic arguments, also described in [San76], is presented at the close of Section 1 as a contrast to those of the subsequent sections. Research supported in part by NSF grant #DMS-9803571.