关键词: 凸体, 宽度, 单形, 体积

Abstract: 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 Vol_n(\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.

Key words: Convex body, Width, Simplex, Volume