Proceedings of the American Mathematical Society· 2026Q1
On the optimality of random partial sphere coverings in high dimensions
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- Q1SCImago
- 2026year
Short summary
In high dimensions ($d o \infty$), the maximum proportion of a unit sphere covered by $N$ geodesic caps (where $N$ grows with $d$ such that $\ln N = o(\sqrt{d})$) asymptotically approaches $1 - e^{-1}$ (approximately 63.2%).
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Key points
- Provides an asymptotically sharp upper bound for the union of $N$ geodesic caps on a unit sphere in $\mathbb{R}^d$.
- The maximum proportion covered approaches $1 - e^{-1}$ as $d \to \infty$ under the condition $\ln N(d) = o(\sqrt{d})$.
- The result relates to the optimality of random polytopes in high dimensions.
- Discusses limitations based on Gaussian surface area bounds.
AI-generated from the title and abstract; the full text is not read.
Abstract
Given $N$ geodesic caps on the unit sphere in $\mathbb{R}^d$, whose total normalized surface area is one, what is the maximal proportion of the sphere that their union can cover? In this work, we provide an asymptotically sharp upper bound for an antipodal partial covering of the sphere by $N=N(d)$ congruent caps in the regime $N(d)\to\infty$ and $\ln N(d)=o(\sqrt d)$, showing that the maximum proportion covered approaches $1 - e^{-1}$ as $d\to\infty$. We discuss the relation of this result to the optimality of random polytopes in high dimensions, the limitations of our technique via the Gaussian surface area bounds of K. Ball and F. Nazarov, and its applications in computer science theory.
The authors' abstract, as published at the source. Proceedings of the American Mathematical Society, 2026 · DOI ↗
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Field: Computer Graphics and Computer-Aided Design
Computer Graphics and Computer-Aided DesignComputer Science