Convex set

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In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex.

Contents

In Euclidean geometry

Let C be a set in a real or complex vector space. C is said to be convex if, for all x and y in C and all t in the interval [0,1], the point

is in C. In other words, every point on the line segment connecting x and y is in C. This implies that a convex set in a real or complex topological vector space is path-connected, thus connected.

A set C is called absolutely convex if it is convex and balanced.

The convex subsets of R (the set of real numbers) are simply the intervals of R. Some examples of convex subsets of Euclidean 2-space are regular polygons and bodies of constant width. Some examples of convex subsets of Euclidean 3-space are the Archimedean solids and the Platonic solids. The Kepler-Poinsot polyhedra are examples of non-convex sets.

Properties

If S is a convex set, for any u_1,u_2,\ldots,u_r in S, and any nonnegative numbers \lambda_1,\lambda_2,\ldots,\lambda_r such that \lambda_1+\lambda_2+\cdots+\lambda_r=1, then the vector \sum_{k=1}^r\lambda_k u_k is in S. A vector of this type is known as a convex combination of u_1,u_2,\ldots,u_r.

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