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Erschienen in:
2011 | OriginalPaper | Buchkapitel
Basic Properties of Convex Sets
verfasst von : Jean Gallier
Erschienen in: Geometric Methods and Applications
Verlag: Springer New York
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Convex sets play a very important role in geometry. In this chapter we state and prove some of the “classics” of convex affine geometry: Carath’eodory’s theorem, Radon’s theorem, Helly’s theorem, and Krein and Millman’s theorem. These theorems share the property that they are easy to state, but they are deep, and their proof, although rather short, requires a lot of creativity. We introduce the notions of separating and supporting hyperplanes, of vertices, and of extreme points. We also define centerpoints and prove their existence.