Face numbers of high-dimensional Poisson zero cells

Let Z d \mathcal Z_d be the zero cell of a alttext="d"> encoding="application/x-tex">d -dimensional, isotropic and stationary Poisson hyperplane tessellation. We study asymptotic behavior expected number alttext="k"> k encoding="application/x-tex">k -dimensional faces , as alttext="d right-arrow normal infinity"> → mathvariant="normal">∞ encoding="application/x-tex">d\to \infty . For example, we show that hyperfaces is asymptotically equivalent to alttext="StartRoot 2 pi slash 3 EndRoot d Superscript 2"> 2 π<!-- π <mml:mo>/ 3 encoding="application/x-tex">\sqrt {2\pi /3}\, d^{3/2} also prove solid angle random cone spanned by vectors are independent uniformly distributed on unit half-sphere in alttext="double-struck R mathvariant="double-struck">R encoding="application/x-tex">\mathbb R^{d} negative −<!-- − \pi ^{-d}</mml:annotation>