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Reduced spherical polygons

Volume 138 / 2015

Marek Lassak Colloquium Mathematicum 138 (2015), 205-216 MSC: Primary 52A55; Secondary 52A10. DOI: 10.4064/cm138-2-5

Abstract

For every hemisphere $K$ supporting a spherically convex body $C$ of the $d$-dimensional sphere $S^d$ we consider the width of $C$ determined by $K$. By the thickness $\varDelta (C)$ of $C$ we mean the minimum of the widths of $C$ over all supporting hemispheres $K$ of $C$. A spherically convex body $R \subset S^d$ is said to be reduced provided $\varDelta (Z) < \varDelta (R)$ for every spherically convex body $Z \subset R$ different from $R$. We characterize reduced spherical polygons on $S^2$. We show that every reduced spherical polygon is of thickness at most $\pi /2$. We also estimate the diameter of reduced spherical polygons in terms of their thickness. Moreover, a few other properties of reduced spherical polygons are given.

Authors

  • Marek LassakInstitute of Mathematics and Physics
    University of Science and Technology
    85-789 Bydgoszcz, Poland
    e-mail

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