Convex Sets: Topological Properties

Abstract

• Why. Convex sets have many useful special topological properties. The challenging concept of recession directions of a convex set has to be mastered: this is needed for work with unbounded convex sets. Here is an example of the use of recession directions: they can turn ‘non-existence’ (of a bound for a convex set or of an optimal solution for a convex optimization problem) into existence (of a recession direction). This gives a certificate for non-existence.

• What. The recession directions of a convex set can be obtained by taking the closure of its homogenization. These directions can be visualized by means of the three models as ‘points at infinity’ of a convex set, lying on ‘the horizon’. All topological properties of convex sets can be summarized by one geometrically intuitive result only: if you adjoin to a convex set A their points at infinity, then the outcome has always the same ‘shape’, whether A is bounded or unbounded: it is a slightly deformed open ball (its relative interior) that is surrounded on all sides by a ‘peel’ (its relative boundary with its points at infinity adjoined). In particular, this is essentially a reduction of unbounded convex sets to the simpler bounded convex sets. This single result implies all standard topological properties of convex sets. For example, there is a very close relation between the relative interior and the closure of a convex set: they can be obtained out of each other by taking in one direction the relative interior and in the other one by the closure. Another example is that the relative boundary of a convex set has a ‘continuity’ property.

Road Map

• Definition 1.5.4 and Fig. 3.1 (recession vector).

• Section 3.2 (list of topological notions).

• Proposition 3.2.1, Definition 1.5.4, Fig. 3.2 (construction of recession vectors by homogenization).

• Corollary 3.2.4 (unboundedness equivalent to existence recession direction).

• Figures 3.3, 3.4, 3.5, 3.6, 3.7 (visualization of recession directions as points on the horizon).

• Theorem 3.5.7 (the shape of bounded and unbounded convex sets).

• Figure 3.10 (the idea behind the continuity of the relative boundary of a convex set).

• Corollary 3.6.2 (all standard topological properties of convex sets).

• Figure 3.11 (warning that the image under a linear function of a closed convex set need not be closed).