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Abstract Consider two half-spaces$$H_1^+$$ and$$H_2^+$$ in$${\mathbb {R}}^{d+1}$$ whose bounding hyperplanes$$H_1$$ and$$H_2$$ are orthogonal and pass through the origin. The intersection$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$ is a spherical convex subset of thed-dimensional unit sphere$${\mathbb {S}}^d$$ , which contains a great subsphere of dimension$$d-2$$ and is called a spherical wedge. Choosenindependent random points uniformly at random on$${\mathbb {S}}_{2,+}^d$$ 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$$\log n$$ . A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$${\mathbb {S}}_{2,+}^d$$ . The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.
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Riesz energy, L2$L^2$ discrepancy, and optimal transport of determinantal point processes on the sphere and the flat torus
Abstract Determinantal point processes exhibit an inherent repulsive behavior, thus providing examples of very evenly distributed point sets on manifolds. In this paper, we study the so‐called harmonic ensemble, defined in terms of Laplace eigenfunctions on the sphere and the flat torus , and the so‐called spherical ensemble on , which originates in random matrix theory. We extend results of Beltrán, Marzo, and Ortega‐Cerdà on the Riesz ‐energy of the harmonic ensemble to the nonsingular regime , and as a corollary find the expected value of the spherical cap discrepancy via the Stolarsky invariance principle. We find the expected value of the discrepancy with respect to axis‐parallel boxes and Euclidean balls of the harmonic ensemble on . We also show that the spherical ensemble and the harmonic ensemble on and with points attain the optimal rate in expectation in the Wasserstein metric , in contrast to independent and identically distributed random points, which are known to lose a factor of .
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- Award ID(s):
- 2202877
- PAR ID:
- 10500211
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Mathematika
- Volume:
- 70
- Issue:
- 2
- ISSN:
- 0025-5793
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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