论文标题
正面锥特别好
Amenable cones are particularly nice
论文作者
论文摘要
舒适性是凸锥的几何特性，它比面部暴露性强，并有助于研究误差范围，以解决锥形可行性问题。在本文中，我们建立了良好的圆锥锥体的众多特性，并研究了凸锥的合理性与其他特性之间的关系，例如良性和投射暴露。 我们表明，封闭凸锥的紧凑型切片的舒适性等同于锥的不合适性，并证明了在交叉点和其他凸操作下保留舒适性的几个结果。然后，可以遵循可调为阳性半矩阵锥片的均匀，双重和其他锥体。 众所周知，投射暴露的锥体是可以正常的，并且可及的锥体很好，但是相反的陈述是开放的问题。我们构建了一个四维锥体的例子，该锥体很好，但不适合。我们还表明，正面的圆锥体在尺寸上呈投射暴露，包括四个。 最后，我们讨论了与我们在这项工作过程中遇到的凸面相关的开放问题，但无法完全解决。
Amenability is a geometric property of convex cones that is stronger than facial exposedness and assists in the study of error bounds for conic feasibility problems. In this paper we establish numerous properties of amenable cones, and investigate the relationships between amenability and other properties of convex cones, such as niceness and projectional exposure. We show that the amenability of a compact slice of a closed convex cone is equivalent to the amenability of the cone, and prove several results on the preservation of amenability under intersections and other convex operations. It then follows that homogeneous, doubly nonnegative and other cones that can be represented as slices of the cone of positive semidefinite matrices, are amenable. It is known that projectionally exposed cones are amenable and that amenable cones are nice, however the converse statements have been open questions. We construct an example of a four-dimensional cone that is nice but not amenable. We also show that amenable cones are projectionally exposed in dimensions up to and including four. We conclude with a discussion on open problems related to facial structure of convex sets that we came across in the course of this work, but were not able to fully resolve.
