Covering a reduced spherical polygon by a disk

Abstract

A spherical convex body R is called reduced if any convex body properly contained in R has thickness smaller than the thickness Δ(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta (R)$$\\end{document} of R. We show that for every reduced spherical convex polygon V there exists its boundary point such that the disk of radius Δ(V)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta (V)$$\\end{document} centered at this point contains V.