分级面提升与自由谱面体的自由极点
Graded face lifts and free extreme points of free spectrahedra
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中文总结 AI 辅助
该研究针对自由谱面体自由极点的量化问题，通过分级面提升技术等构造了相关自由极点，得到多面体上最小矩阵凸集成为自由谱面体的障碍等结论。
中文摘要 AI 辅助
自由谱面体是自由线性矩阵不等式 $L_A(X) = I-A_1 \otimes X_1 - \dots - A_g \otimes X_g \succeq 0$ 的矩阵解集合。在这个无维度的框架中，自由极点扮演着经典极点的角色，具体而言，每个有界实自由谱面体都是其自由极点的矩阵凸包，从这个定性意义上来说，自由谱面体的自由极点十分丰富。然而，这种丰富性的量化研究一直难以实现，特别是除单纯形之外，目前尚不清楚是否存在具有有限个自由极点的有界实自由谱面体。拥有有限个自由极点的一个必要条件是，由 $L_A(x) \succeq 0$ 定义的经典谱面体是一个多面体。我们通过两种构造强化了这一必要条件：第一，我们对边数至少为4的多边形上的最大矩阵凸集的第二层，提供了一个无限族自由极点的几何构造；第二，我们为一般有界实自由谱面体开发了分级面提升技术，该技术允许我们从分级面提升的自由极点构造出完整谱面体的自由极点。作为推论，我们得到了多面体上最小矩阵凸集成为自由谱面体的障碍，并表明多面体上最大矩阵凸集的大量标量极点几何可由更高层的自由极点捕获。
英文摘要
A free spectrahedron is the matricial solution set of a free linear matrix inequality $L_A(X) = I-A_1 \otimes X_1 - \dots - A_g \otimes X_g \succeq 0$. In this dimension-free setting, free extreme points play the role of classical extreme points. In particular, every bounded real free spectrahedron is the matrix convex hull of its free extreme points. In this qualitative sense, free extreme points of free spectrahedra are abundant. However, quantifications of this abundance have remained elusive. In particular, outside simplices, it is not known whether there exist bounded real free spectrahedra that have finitely many free extreme points. A necessary condition for having finitely many free extreme points is that the classical spectrahedron defined by $L_A(x) \succeq 0$ is a polytope. We strengthen this necessary condition through two constructions. First, we provide a geometric construction of an infinite family of free extreme points at the second level of the maximal matrix convex set over a polygon with at least four sides. Second, we develop a graded face lifting technique for general bounded real free spectrahedra, which allows us to construct free extreme points of the full spectrahedron from free extreme points of the graded face lift. As corollaries, we obtain obstructions to minimal matrix convex sets over polytopes being free spectrahedra and show that much of the scalar extreme-point geometry of maximal matrix convex sets over polytopes can be captured by higher-level free extreme points.
