We prove an explicit combinatorial formula for the expected number of faces of the zero polytope of the homogeneous and isotropic Poisson hyperplane tessellation in $\mathbb R^d$. The expected $f$-vector is expressed through the coefficients of the polynomial $$ (1+ (d-1)^2x^2) (1+(d-3)^2 x^2) (1+(d-5)^2 x^2) \ldots. $$ Also, we compute explicitly the expected $f$-vector and the expected volume of the spherical convex hull of $n$ random points sampled uniformly and independently from the $d$-dimensional half-sphere. In the case when $n=d+2$, we compute the probability that this spherical convex hull is a spherical simplex, thus solving an analogue of the Sylvester four-point problem on the half-sphere.

中文翻译：

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半球体中泊松零多面体和随机凸包的预期 f 向量

我们证明了 $\mathbb R^d$ 中均匀和各向同性泊松超平面细分的零多胞体的预期面数的显式组合公式。预期的 $f$-vector 通过多项式 $$ (1+ (d-1)^2x^2) (1+(d-3)^2 x^2) (1+(d- 5)^2 x^2) \ldots。$$ 此外，我们明确地计算了 $n$ 随机点的预期 $f$ 向量和球形凸包的预期体积，这些随机点从 $d$ 维半球均匀采样并独立。在 $n=d+2$ 的情况下，我们计算这个球形凸包是球形单纯形的概率，从而解决半球上的 Sylvester 四点问题的模拟。