**1**

Basic convexity

**1.1 Convex sets and combinations**

A set is *convex* if together with any two points *x, y* it contains the segment [*x, y*], thus if

Examples of convex sets are obvious; but observe also that *B*_{0}(*z, ρ*) ∪ *A* is convex if *A* is an arbitrary subset of the boundary of the open ball *B*_{0}(*z, ρ*). As immediate consequences of the definition we note that intersections of convex sets are convex, affine images and pre-images of convex sets are convex, and if *A, B* are convex, then *A* + *B* and *λA* are convex.

**Remark 1.1.1** For and *λ, μ* > 0 one trivially ...

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