Prove that K ⊂ Rn is a convex set if and only if every convex linear combination of elements in K also belongs to K. Proof:(⇒) we assume that S consists m ...
Proof: Let Kα}α∈A be a family of convex sets, and let K := ∩α∈AKα. Then, for any x, y ∈ K by definition of the intersection of a family of sets, x, y ∈ Kα ...
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. Given an ...
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset ...
We can prove convexity preservation under intersection and affine transforms trivially from the definition of a convex set. For perspective transforms we show ...