Journal of Convex Analysis 33 (2026), No. 1&2, 415--420
Copyright Heldermann Verlag 2026
Approximation of Spherical Convex Bodies of Constant Width π/2
Huhe Han
College of Science, Northwest Agriculture and Forestry University, Xianyang, Shaanxi, China
han-huhe@nwafu.edu.cn
[Abstract-pdf]
Let $\mathbb{S}^2$ be the unit sphere in $\mathbb{R}^3$ and let $C\subset \mathbb{S}^2$ be a spherical convex body of constant width $\tau$. It is known that\\ (i) if $\tau<\pi/2$ then for any $\varepsilon>0$ there exists a spherical convex body $C_\varepsilon$ of constant width $\tau$ whose boundary consists only of arcs of circles of radius $\tau$ such that the Hausdorff distance between $C$ and $C_\varepsilon$ is at most $\varepsilon$;\\ (ii) if $\tau>\pi/2$ then for any $\varepsilon>0$ there exists a spherical convex body $C_\varepsilon$ of constant width $\tau$ whose boundary consists only of arcs of circles of radius $\tau-\frac{\pi}{2}$ and great circle arcs such that the Hausdorff distance between $C$ and $C_\varepsilon$ is at most $\varepsilon$.\\ In this paper, we present an approximation of the remaining case $\tau=\pi/2$, that is, if $\tau=\pi/2$ then for any $\varepsilon>0$ there exists a spherical polygon $\mathcal{P}_\varepsilon$ of constant width $\pi/2$ such that the Hausdorff distance between $C$ and $\mathcal{P}_\varepsilon$ is at most $\varepsilon$.
Keywords: Constant width, approximation, spherical polytgon, Hausdorff distance.
MSC: 52A55.
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