论文标题
正面跨越集的简单和圆锥形分解
Simplicial and Conical Decomposition of Positively Spanning Sets
论文作者
论文摘要
我们研究了一组$ x $的分解，该$ x $肯定地跨越了欧几里得空间$ \ mathbb {r}^{d} $的一组最小的正底基，我们将简单命名为简单，并将其称为最大集合，并将其肯定地跨度跨度尖端尖头锥，即带有一个Apex的圆锥。对于任何集合$ x $，令$ \ natercal {s}（x）$表示$ x $的单纯键子集，然后让$ \ ell（x）$表示$ x $的线性船体。当且仅当每个子集$ y \ y \ subset x $和每个单纯子$ s \ in \ mathcal {s}（x）$，$ \ ell（y）\ ell（y）\ cap \ ell（s）= \ ell（y \ cap s）$时，设置$ x $被说满足了分解条件。我们证明$ x $是且仅当它是大多数D简单的结合时，而$ x $才能满足分解条件的积极基础。在这种情况下，$ x $包含线性基础$ b $，以使$ \ mathcal {s}（x）$具有$ b $中的每个单纯词，除了一个恰好是一个共同的一个元素。我们表明，对于积极跨越$ \ mathbb {r}^{d} $的集合，$ x $的子基集形式为布尔晶格，可以将其嵌入$ 2^{\ Mathcal {s s}（x）（x）} $中，并带有isomorphy。我们的第二个主要结果取决于前者，如下所示。有限的集合$ x \ subset \ mathbb {r}^{d} \ setMinus \ {0 \} $可以写为最多$ 2^{d} $最大设置的结合。当且仅当$ x $是一个十字架时，即从$ \ mathbb {r}^{d} $的线性基础得出的1个简化的联合时，不平等却急剧保持。我们还表明，当其中两个相交的交叉点不跨越一组完整尺寸时，可能会有$ 2^{d} $最大子集的最大子集的最大子集。
We investigate the decomposition of a set $X$, which positively spans the Euclidean space $\mathbb{R}^{d}$ into a set of minimal positive bases, we call simplices, and into maximal sets positively spanning pointed cones, i.e. cones with exactly one apex. For any set $X$, let $\mathcal{S}(X)$ denote the set of simplex subsets of $X$, and let $\ell(X)$ denote the linear hull of $X$. The set $X$ is said to fulfill the factorisation condition if and only if for each subset $Y\subset X$ and each simplex $S\in\mathcal{S}(X)$, $\ell(Y)\cap\ell(S) = \ell(Y\cap S)$. We demonstrate that $X$ is a positive basis if and only if it is the union of most d simplices, and $X$ satisfies the factorization condition. In this case, $X$ contains a linear basis $B$ such that each simplex in $\mathcal{S}(X)$ has with $B$, all but one exactly one element in common. We show that for sets positively spanning $\mathbb{R}^{d}$, the set of subbases of $X$ forms a boolean lattice, which can be embedded into the set $2^{\mathcal{S}(X)}$, with isomorphy for positive bases. Our second main result depending on the former is as follows. A finite set $X\subset\mathbb{R}^{d}\setminus\{0\}$ can be written as the union of at most $2^{d}$ maximal sets spanning pointed cones, which, if $X$ is a positive basis, are tantamount to frames of the cones. The inequality holds sharply if and only if $X$ is a cross, that is, a union of 1-simplices derived from a linear basis of $\mathbb{R}^{d}$. We also show that there can be at the most $2^{d}$ maximal subsets of $X$ spanning pointed cones, when intersections of two of them do not span a set of full dimension.
