Problems on polygons and Bonnesen-type inequalities

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 positive Gauss curvature) and the hyperbolic plane (having constant negative Gauss curvature). These generalized inequalities each converge to the classical Bonnesen-type inequality for the Euclidean plane as the curvature approaches zero.

Santalo关于强Bonnesen不等式积分几何证明的一个新技巧

A new trick for for Santalo's integral geometric proof of the strong Bonnesen inequality

本文通过引入“平均技巧”平衡被积函数的正负区域，证明了Bonnesen泛函在最小宽度圆环内外半径区间上非负，从而为强Bonnesen不等式提供了新的积分几何证明，并应用于曲线缩短流的最终形状分析。

Affine isoperimetric inequalities for piecewise linear surfaces

This paper considers affine analogues of the isoperimetric inequality in the sense of piecewise linear topology. Given a closed polygon P embedded in R^d having n edges, we give upper and lower bounds for the minimal number of triangles needed to forma triangulated embedded orientable surface in R^d having P as its geometric boundary. The most interesting case is dimension dimension 3, where we give an upper bound of 7 n^2 triangles, and a lower bound for some polygons P that require at least 1/2 n^2 triangles. In dimension 2 and dimensions 5 and above one needs only O(n) triangles. The case o…

Metric inequalities for polygons

Let $A_1,A_2,...,A_n$ be the vertices of a polygon with unit perimeter, that is $\sum_{i=1}^n |A_i A_{i+1}|=1$. We derive various tight estimates on the minimum and maximum values of the sum of pairwise distances, and respectively sum of pairwise squared distances among its vertices. In most cases such estimates on these sums in the literature were known only for convex polygons. In the second part, we turn to a problem of Braß regarding the maximum perimeter of a simple $n$-gon ($n$ odd) contained in a disk of unit radius. The problem was solved by Audet et al. \cite{AHM09b}, who gave an exac…