Consider two halfspaces
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Abstract and$$H_1^+$$ ${H}_{1}^{+}$ in$$H_2^+$$ ${H}_{2}^{+}$ whose bounding hyperplanes$${\mathbb {R}}^{d+1}$$ ${R}^{d+1}$ and$$H_1$$ ${H}_{1}$ are orthogonal and pass through the origin. The intersection$$H_2$$ ${H}_{2}$ is a spherical convex subset of the$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$ ${S}_{2,+}^{d}:={S}^{d}\cap {H}_{1}^{+}\cap {H}_{2}^{+}$d dimensional unit sphere , which contains a great subsphere of dimension$${\mathbb {S}}^d$$ ${S}^{d}$ and is called a spherical wedge. Choose$$d2$$ $d2$n independent random points uniformly at random on and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of$${\mathbb {S}}_{2,+}^d$$ ${S}_{2,+}^{d}$ . A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$$\log n$$ $logn$ . The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a halfsphere.$${\mathbb {S}}_{2,+}^d$$ ${S}_{2,+}^{d}$ 
Besau, Florian ; Hoehner, Steven ; Kur, Gil ( , International Mathematics Research Notices)
Abstract We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced by the Wills functional and apply it to derive asymptotically sharp bounds for approximating the ball in high dimensions. Remarkably, it turns out that there is a polytope that is almost optimal with respect to all intrinsic volumes simultaneously, up to absolute constants.
Finally, we establish asymptotic formulas for the best approximation of smooth convex bodies by polytopes with respect to a distance induced by dual volumes, which originate from Lutwak’s dual Brunn–Minkowski theory.