The beta polytope is the convex hull of n i.i.d. random points distributed in the unit ball of according to a density proportional to if (in particular, corresponds to the uniform distribution in the ball), or uniformly on the unit sphere if . We show that the expected normalized volumes of high‐dimensional beta polytopes exhibit a phase transition and we describe its shape. We derive analogous results for the intrinsic volumes of beta polytopes and, when , their number of vertices.

中文翻译：

---

高维随机多态性的体积的相变

所述β-多面体是凸包Ñ IID分布在单位球上的随机点根据密度成比例，如果（特别是，对应于球的均匀分布），或均匀地在单位球面上如果。我们表明，预期的高维β多表位归一化体积显示出相变，并描述了其形状。我们得出β多表位的内在体积及其顶点数的相似结果。