Inequalities for Widths of Convex Bodies with Applications
凸体的宽度不等式及应用

In this paper the authors establish the following inverse inequality of Yang-Zhang's inequality for the width of a simplex: Let $\Omega$ be an n-dimensional simplex with volume Voln(\Omega)$,width $w(\Omega)$, and facet areas $S_1,S_2,\cdots,S_{n+1}$ respectively, then$$w(\Omega)\ge r_n\cdot\frac{{{\rm Vol}_n}(\Omega)}{\displaystyle\max_{1\le i\le n+1}(S_i)},$$where $$\gamma_n=\left\{\begin{array}{cl}\disp \frac{2n}{n+1}, & \qquad {\rm for~ odd}~~ n;\\2, & \qquad {\rm for~ even}~~ n.\end{array} \right.$$As applications, the authors show some inequalities for orthogonal projections and sections of convex bodies.