Convex Sets

Abstract

The theory of convex polytopes, and more generally the theory of convex sets, belongs to the subject of affine geometry. In a sense, the right framework for studying convex sets is the notion of a Euclidean space, i.e. a finite-dimensional real affine space whose underlying linear space is equipped with an inner product. However, there is no essential loss of generality in working only with the more concrete spaces ℝd; therefore, everything will take place in ℝd. We will assume that the reader is familiar with the standard linear theory of ℝd, including such notions as subspaces, linear independence, dimension, and linear mappings. We also assume familiarity with the standard inner product <·, ·> of ℝd, including the induced norm ∥ ∥, and elementary topological notions such as the interior int M, the closure cl M, and the boundary bd M of a subset M of ℝd.KeywordsLinear SubspaceConvex CombinationAffine MappingAffine SpaceSupporting HyperplaneThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.