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A Study on Spherical Convexity  

Jo, Kyeonghee (Division of Liberal Arts and Sciences, Mokpo National Maritime Univ.)
Publication Information
Journal for History of Mathematics / v.29, no.6, 2016 , pp. 335-351 More about this Journal
Spherical convexity may be defined in different ways. It depends on which statement we take as a definition among several statements that can be all used as a definition of convexity of subsets in an affine space. In this article, we consider this question from various perspectives. We compare several different definitions of spherical convexity which are found in mathematical papers. In particular, we focus our discussion on the definitions of J. P. $Benz{\acute{e}}cri$ and N. H. Kuiper who built a solid foundation for theory of convex bodies and convex affine(projective) structures on manifolds.
convexity; convex domains; spherical convexity;
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1 G. AUBRUN, M. FRADELIZI, Two-point symmetrization and convexity, Arch. Math. 82 (2004), 282-288.   DOI
2 J. P. BENZECRI, Sur les varietes localement affines et projectives, Bull. Soc. Math. France 88 (1960), 229-332.
3 F. J. COBOS et al, The width of a convex set on the sphere, Proceeding of the 9th Canadian Conference on Computational Geometry, Kingston, Ontarlo, Canada, Aug. 11-14, 1997.
4 D. DEKKER, Convex regions in projective space, The Amer. Math. Monthly 62(6) (1955), 430-431.   DOI
5 O. P. FERREIRA, A. N. IUSEM, S. Z. NEMETH, Projections onto convex sets on the sphere, Jour. Global Optimization 57(3) (2013), 663-676.   DOI
6 J. de GROOT, H. de VRIES, Convex sets in projective space, Compositio Mathematica 13 (1956-1958), 113-118.
7 B. P. HAALMEYER, Bijdragen tot de theorie der elementairoppervlakken, Amsterdam, 1917.
8 L. HORMANDER, Notion of convexity, Mordern Birkhauser Classics, 1994.
9 H. KNESER, Eine Erweiterung des Begriffes "konvexe Korper", Math. Ann. 82 (1921), 287-296.   DOI
10 N. H. KUIPER, On convex locally projective spaces, Convegno Intern. Geom. Diff. Italy, 1953, 200-213.
11 K. MENGER, Urtersuchugen uber allgemeine Metrik, Math. Ann. 100 (1928), 75-163.   DOI
12 D. MINDA, The hyperbolic metric and bloch constants for spherically convex regions, Complex Variables 5 (1986), 127-140.   DOI
13 E. STEINITZ, Bedingt konvergente Reihen und konvexe Systeme. Teil III, J. Reine Anrgew. Math. 146 (1916), 1-52.
14 T. TODDA, Convex sets in a real projective space and its application to computational geometry, manuscript.
15 T. TODDA, Multi-convex sets in real projective spaces and their duality, manuscript.
16 J. H. C. WHITEHEAD, Convex regions in the geometry of paths, Differential geometry: The Mathematical Works of J. H. C. Whitehead, (2014), 223-232.
17 R. SCHNEIDER, Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, 1993.