Geodesic
Two Applications of Topology to Convex Geometry
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology and set theory, Tome 247 (2004), pp. 182-185

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The purpose of this paper is to prove two theorems of convex geometry using the techniques of topology. The first theorem states that if, for a strictly convex body $K$, one may choose continuously a centrally symmetric section, then $K$ must be centrally symmetric. The second theorem states that if every section of a three-dimensional convex body $K$ through the origin has an axis of symmetry, then there is a section of $K$ through the origin which is a disk. 
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     author = {L. Montejano},
     title = {Two {Applications} of {Topology} to {Convex} {Geometry}},
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L. Montejano. Two Applications of Topology to Convex Geometry. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology and set theory, Tome 247 (2004), pp. 182-185. http://geodesic.mathdoc.fr/item/TM_2004_247_a11/