凸体的穿透平移和同形异义
Piercing translates and homothets of a convex body
摘要：
根据Grünbaum的一个经典结果，凸体$C$在$\RR^d$中的任意两交平移或同素异形族$\F$的横截数$τ(\F)$都有界于$d$的一个函数.用$α(C)$表示（分别$β(C)$）在所有的平移族$\F$上的横截数$τ(\F)$与填充数$ν(\F)$之比的上确界（分别为同形异义词）的凸体$C$在$\RR^d$。Kim等人最近证明了对于$\RR^d$中的任何凸体$C$，$α(C)$都受$d$的函数的约束，并给出了对于$\RR^d$中的凸体$C$，$α(C)$的第一界和对于平面中的凸体$C$，$β(C)$的第一界。在这里，我们表明，$β(C)$也有界的函数$d$的任何凸体$C$在$\RR^d$，并提出新的或改进的界限$α(C)$和$β(C)$的各种凸体$C$在$\RR^d$的所有维度$d$。我们的技术探索有趣的不等式连接覆盖和包装密度的凸体。我们的方法获得上界是建设性的，并导致有效的常数因子近似算法找到一个最小的基数点集，穿透一组的翻译或同胚的凸体。
英文摘要：
According to a classical result of Grünbaum, the transversal number $τ(\F)$ of any family $\F$ of pairwise-intersecting translates or homothets of a convex body $C$ in $\RR^d$ is bounded by a function of $d$. Denote by $α(C)$ (resp. $β(C)$) the supremum of the ratio of the transversal number $τ(\F)$ to the packing number $ν(\F)$ over all families $\F$ of translates (resp. homothets) of a convex body $C$ in $\RR^d$. Kim et al. recently showed that $α(C)$ is bounded by a function of $d$ for any convex body $C$ in $\RR^d$, and gave the first bounds on $α(C)$ for convex bodies $C$ in $\RR^d$ and on $β(C)$ for convex bodies $C$ in the plane. Here we show that $β(C)$ is also bounded by a function of $d$ for any convex body $C$ in $\RR^d$, and present new or improved bounds on both $α(C)$ and $β(C)$ for various convex bodies $C$ in $\RR^d$ for all dimensions $d$. Our techniques explore interesting inequalities linking the covering and packing densities of a convex body. Our methods for obtaining upper bounds are constructive and lead to efficient constant-factor approximation algorithms for finding a minimum-cardinality point set that pierces a set of translates or homothets of a convex body.
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