关于凸体边界的带

一致凸赋范空间紧子集中的区域图

Zone diagrams in compact subsets of uniformly convex normed spaces

区域图是计算几何中出现的一个相对较新的概念，与Voronoi图有关。形式上，它是某个映射的一个不动点，它的唯一性和存在性事先都不明显。它已经被几位作者研究过，从T。Asano，J. Matousek和T.德山，谁认为欧几里德平面与单点，并证明了存在性和唯一性的区域图。本文证明了一致凸赋范空间的紧凸子集中包含的多个两两不相交紧点的区域图的存在性，只要这些点或凸子集满足一定的温和条件。证明是基于Schauder不动点定理，Curtis-肖里定理关于希尔伯特立方体，和最近的结果有关的表征Voronoi细胞作为一个集合的线段和它们的几何稳定性相对于小的变化，相应的网站。一路上，我们得到的DOM映射的连续性，以及有趣的，显然是新的性质的Voronoi细胞。

Active Learning a Convex Body in Low Dimensions

Consider a set $P \subseteq \Re^d$ of $n$ points, and a convex body $C$ provided via a separation oracle. The task at hand is to decide for each point of $P$ if it is in $C$ using the fewest number of oracle queries. We show that one can solve this problem in two and three dimensions using $O( h(P) \log n)$ queries, where $h(P)$ is the largest subset of points of $P$ in convex position. Furthermore, we show that in two dimensions one can solve this problem using $O( v(P,C) \log^2 n )$ oracle queries, where $v(P, C)$ is a lower bound on the minimum number of queries that any algorithm for this …

凸体的穿透平移和同形异义

Piercing translates and homothets of a convex body

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。

Economical Convex Coverings and Applications

Coverings of convex bodies have emerged as a central component in the design of efficient solutions to approximation problems involving convex bodies. Intuitively, given a convex body $K$ and $ε> 0$, a covering is a collection of convex bodies whose union covers $K$ such that a constant factor expansion of each body lies within an $ε$ expansion of $K$. Coverings have been employed in many applications, such as approximations for diameter, width, and $ε$-kernels of point sets, approximate nearest neighbor searching, polytope approximations, and approximations to the Closest Vector Problem (CVP)…

On the streaming complexity of fundamental geometric problems

In this paper, we focus on lower bounds and algorithms for some basic geometric problems in the one-pass (insertion only) streaming model. The problems considered are grouped into three categories: (i) Klee's measure (ii) Convex body approximation, geometric query, and (iii) Discrepancy Klee's measure is the problem of finding the area of the union of hyperrectangles. Under convex body approximation, we consider the problems of convex hull, convex body approximation, linear programming in fixed dimensions. The results for convex body approximation implies a property testing type result to find…

学习凸体很难

Learning convex bodies is hard

我们表明，学习凸体$\RR^d$，从身体的随机样本，需要$2^{Ω(\sqrt{d/\eps})}$样本。通过学习凸体，我们的意思是找到一个与输入体至多具有$\eps$相对对称差的集合。为了证明下界，我们构造了一个很难学习的凸体族。我们构造这个族是非常简单的，并且基于纠错码。