The Bonnesen-Type Inequalities in a Plane of Constant Curvature

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]). …”

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]). …”

“…Bonnesen, Ren and Zhou obtained Bonnesen-style inequalities by kinematic formulas and the containment measure in integral geometry (see [30,33,36,37,43,44,45,46,47,48]). Other researchers obtained Bonnesen-style inequalities by using the approaches in differential geometry and analysis.…”

On the Isoperimetric Deficit Upper Limit

“…An inequality of type (3) is called a Bonnesen style inequality. See [2], [3], [4], [6], [10], [16], [19], [21], [24], [25], [34] and [33] for more detailed references.…”

“…One can find some simplified and beautiful proofs that lead to generalizations of higher dimensions and applications to other branches of mathematics (cf. [1], [6], [10], [11]- [17], [16], [20], [21], [24]- [25], [27], [29], [30], [31], [32], [34]- [35], [37], [39]). …”