有限集的$2+1$凸包
The $2+1$ convex hull of a finite set
摘要：
我们研究$\mathbb{R}^2\oplus\mathbb{R}$ -分别凸包的有限集的点在$\mathbb{R}^3$，在KirchheimMullerSverak 2003。当$\mathbb{R}^3$与矩阵的某些子集相同时，我们称之为$2+1$凸性的凸性概念对应于秩一凸凸性或拟凸性。本文引入$2+1$复形，推广了$T_n$结构，定义了集合的$2+1$复形凸包，并证明了它是$2+1$凸包的内逼近.我们还考虑了基于秩凸局部性定理的$2+1$凸性的外逼近，通过迭代地截断“$D$ -棱柱”。对于许多有限集，该过程在有限数目的步骤中达到“$2+1$ $K$ -复形”，从而计算$2+1$凸包。我们展示的例子有限集，这一过程不达到$2+1$凸包在有限数量的步骤，但我们表明，总是有一个序列的外近似建立与$D$ -棱镜收敛到一个$2+1$ $K$ -复杂。我们的结论是，$K^{rc}$总是一个“$2+1$ $K$ -复合体”，这有有趣的后果。
英文摘要：
We study $\mathbb{R}^2\oplus\mathbb{R}$-separately convex hulls of finite sets of points in $\mathbb{R}^3$, as in KirchheimMullerSverak2003. This notion of convexity, which we call $2+1$ convexity, corresponds to rank-one convex convexity, or quasiconvexity, when $\mathbb{R}^3$ is identified with certain subsets of matrices. We introduce '$2+1$ complexes', which generalize $T_n$ constructions, define the '$2+1$-complex convex hull of a set', and prove that it is an inner approximation to the $2+1$ convex hull. We also consider outer approximations to $2+1$ convexity based in the locality theorem of rank convexity, by iteratively chopping off '$D$-prisms'. For many finite sets, this procedure reaches a '$2+1$ $K$-complex' in a finite number of steps, and thus computes the $2+1$ convex hull. We show examples of finite sets for which this procedure does not reach the $2+1$ convex hull in a finite number of steps, but we show that there is always a sequence of outer approximations built with $D$-prisms that converges to a $2+1$ $K$-complex. We conclude that $K^{rc}$ is always a '$2+1$ $K$-complex', which has interesting consequences.
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