具有相同回收锥的凸集的极小对
Minimal pairs of convex sets which share a recession cone
AI总结：
该文研究具有相同回收锥的无界凸集极小对，证明相关性质，给出二维情形的计算公式与极小性判据，推广多面体和项判据，并展示其在dc-函数极小表示中的应用。
摘要：
Robinson引入了具有相同回收锥的凸集对的商空间。本文研究了无界凸集的极小对，即Robinson空间中元素的极小表示。证明了具有平移性质的极小对是可约的。针对二维集对的情形，给出了等价极小对的计算公式，提出了集对极小性的判据，并证明了所有极小对的可约性。将Shephard–Weil–Schneider关于紧凸集的多面体和项的判据推广到无界凸集。展示了无界凸集极小对在Hartman的dc-函数极小表示中的应用。文中还给出了三维集极小对的示例。
英文摘要：
Robinson introduced a quotient space of pairs of convex sets which share their recession cone. In this paper minimal pairs of unbounded convex sets, i.e. minimal representations of elements of Robinson's spaces are investigated. The fact that a minimal pair having property of translation is reduced is proved. In the case of pairs of two-dimensional sets a formula for an equivalent minimal pair is given, a criterion of minimality of a pair of sets is presented and reducibility of all minimal pairs is proved. Shephard--Weil--Schneider's criterion for polytopal summand of a compact convex set is generalized to unbounded convex sets. An application of minimal pairs of unbounded convex sets to Hartman's minimal representation of dc-functions is shown. Examples of minimal pairs of three-dimensional sets are given.
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