on some discrete Bonnesen-style isoperimetric inequalities

This article deals with the sharp discrete isoperimetric inequalities in analysis and geometry for planar convex polygons. First, the analytic isoperimetric inequalities based on Schur convex function are established. In the wake of the analytic isoperimetric inequalities, Bonnesen-style isoperimetric inequalities and inverse Bonnesen-style inequalities for the planar convex polygons are obtained.Comment: 17 pages, 2 figure

SL(n) Contravariant Matrix-Valued Valuations on Polytopes

All $\textrm{SL}(n)$ contravariant matrix-valued valuations on polytopes in $\mathbb{R}^n$ are completely classified without any continuity assumptions. Moreover, the symmetry assumption of matrices is removed. The general Lutwak-Yang-Zhang matrix turns out to be the only such valuation if $n\geq 4$, while a new function shows up in dimension three. In dimension two, the classification corresponds to the known case of $\textrm{SL}(2)$ equivariant matrix-valued valuations

A new proof of the Wulff-Gage isoperimetric inequality and its applications

A new proof of the Wulff-Gage isoperimetric inequality for origin-symmetric convex bodies is provided. As its applications, we prove the uniqueness of log-Minkowski problem and a new proof of the log-Minkowski inequality of curvature entropy for origin-symmetric convex bodies of $C^{2}$ boundaries in $\mathbb R^{2}$ is given

Some Sharp Chernoff type inequalities

Two sharp Chernoff type inequalities are obtained for star body in $\mathbb{R}^2$, one of which is an extension of the dual Chernoff-Ou-Pan inequality, and the other is the reverse Chernoff type inequality. Furthermore, we establish a generalized dual symmetric mixed Chernoff inequality for two planar star bodies. As a direct consequence, a new proof of the the dual symmetric mixed isoperimetric inequality is presented