Convex Hulls, Oracles, and Homology

Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

Given $n$ distinct points $\mathbf{x}_1, \ldots, \mathbf{x}_n$ in $\mathbb{R}^d$, let $K$ denote their convex hull, which we assume to be $d$-dimensional, and $B = \partial K $ its $(d-1)$-dimensional boundary. We construct an explicit one-parameter family of continuous maps $\mathbf{f}_{\varepsilon} \colon \mathbb{S}^{d-1} \to K$ which, for $\varepsilon > 0$, are defined on the $(d-1)$-dimensional sphere and have the property that the images $\mathbf{f}_{\varepsilon}(\mathbb{S}^{d-1})$ are codimension $1$ submanifolds contained in the interior of $K$. Moreover, as the parameter $\varepsilon$ …

The persistent homology of cyclic graphs

We give an $O(n^2(k+\log n))$ algorithm for computing the $k$-dimensional persistent homology of a filtration of clique complexes of cyclic graphs on $n$ vertices. This is nearly quadratic in the number of vertices $n$, and therefore a large improvement upon the traditional persistent homology algorithm, which is cubic in the number of simplices of dimension at most $k+1$, and hence of running time $O(n^{3(k+2)})$ in the number of vertices $n$. Our algorithm applies, for example, to Vietoris--Rips complexes of points sampled from a curve in $\mathbb{R}^d$ when the scale is bounded depending on…

锥与约束彩色卡拉西奥多里定理

Cone and constrained colorful Carathéodory Theorems

本文在拓扑学上扩展霍尔姆森等人的方法，研究凸锥取代凸包及横截约束非由拟阵编码时的彩色卡拉西奥多里型结果，通过分析单纯复形同调群，如定向拟阵支撑复形，实现经典特弗伯格定理新扩展及方法精细应用。