凸体的宽度不等式及应用

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Inequalities for Widths of Convex Bodies with Applications

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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 (Ω) \geq r_{n} \cdot \frac{{V o l}_{n} (Ω)}{max_{1 \leq i \leq n + 1} (S_{i})},

$w(\Omega)\ge r_n\cdot\frac{{{\rm Vol}_n}(\Omega)}{\displaystyle\max_{1\le i\le n+1}(S_i)},$
where

γ_{n} = {\begin{array}{cl} \disp \frac{2 n}{n + 1}, & f o r o d d n; \\ 2, & f o r e v e n n . \end{array}

$\gamma_n=\left\{\begin{array}{cl} \disp \frac{2n}{n+1}, & \qquad {\rm for~ odd}~~ n;\\ 2, & \qquad {\rm for~ even}~~ n. \end{array} \right.$
As applications, the authors show some inequalities for orthogonal projections and sections of convex bodies.

Key words: Convex body, Width, Simplex, Volume

CLC Number:

52A40

Yuan Shufeng; Ke Rui; Leng Gangsong. Inequalities for Widths of Convex Bodies with Applications[J].Acta mathematica scientia,Series A, 2007, 27(4): 660-664.

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