论文标题
对巴拉尼关于多面体数量的问题的积极答案
A Positive Answer to Bárány's Question on Face Numbers of Polytopes
论文作者
论文摘要
尽管简单和简单多型的面部矢量充分表征，但对一般多型的脸部数量却很少。在1997年左右，巴拉尼（Bárány）问是否在所有凸$ d $ -polytopes $ p $和所有$ 0 \ leq k \ leq d-1 $，$ f_k（p）\ geq \ min \ min \ f_0（p），f_ {d-1}（d-1}（p）（p）\} $。我们在肯定中回答了Bárány的问题，并证明了一个更强有力的陈述：对于所有凸出$ d $ -polytopes $ p $和所有$ 0 \ leq k \ leq d-1 $，\ [\ frac {f_k（p）} \ frac {d} {2} \ rceil \ select k} + {\ lfloor \ frac {d} {2} {2} \ rfloor \ select k} \ biggr]，\ qquad \ qquad \ frac \ frac {f_k（p）} \ frac {1} {2} \ biggl [{\ lceil \ frac {d} {2} {2} \ rceil \ select d-k-1} + {\ lfloor \ frac {d} {d} {2} {2} {2} {2} {2} \ rfloor \ rfloor \ osece d-k-k-1} \ biggr]。 \]在前者中，当$ k = 0 $或$ k = 1 $和$ p $很简单时，平等恰好保持。在后者中，当$ k = d-1 $或$ k = d-2 $和$ p $时，平等恰好保持。
Despite a full characterization of the face vectors of simple and simplicial polytopes, the face numbers of general polytopes are poorly understood. Around 1997, Bárány asked whether for all convex $d$-polytopes $P$ and all $0 \leq k \leq d-1$, $f_k(P) \geq \min\{f_0(P), f_{d-1}(P)\}$. We answer Bárány's question in the affirmative and prove a stronger statement: for all convex $d$-polytopes $P$ and all $0 \leq k \leq d-1$, \[ \frac{f_k(P)}{f_0(P)} \geq \frac{1}{2}\biggl[{\lceil \frac{d}{2} \rceil \choose k} + {\lfloor \frac{d}{2} \rfloor \choose k}\biggr], \qquad \frac{f_k(P)}{f_{d-1}(P)} \geq \frac{1}{2}\biggl[{\lceil \frac{d}{2} \rceil \choose d-k-1} + {\lfloor \frac{d}{2} \rfloor \choose d-k-1}\biggr]. \] In the former, equality holds precisely when $k=0$ or when $k=1$ and $P$ is simple. In the latter, equality holds precisely when $k=d-1$ or when $k=d-2$ and $P$ is simplicial.
