Knowlpedia

Affine images and preimages of convex sets are convex

Affine maps preserve convexity under both images and inverse images

Proposition. Let B:X→YB:X\to Y be an .

  1. If Ω⊂X\Omega\subset X is , then B(Ω)⊂YB(\Omega)\subset Y is convex.
  2. If Θ⊂Y\Theta\subset Y is convex, then the preimage B−1(Θ)={x∈X:B(x)∈Θ}B^{-1}(\Theta)=\{x\in X:B(x)\in\Theta\} is convex in XX.
Remarks

Context. This is the main mechanism for generating new convex sets from old ones: apply an affine change of coordinates, or pull back convex constraints.

Proof sketch. For (1), take u=B(x)u=B(x) and v=B(y)v=B(y) with x,y∈Ωx,y\in\Omega and use the defining identity for affine maps:

λu+(1−λ)v=B(λx+(1−λ)y)∈B(Ω).\lambda u+(1-\lambda)v = B(\lambda x+(1-\lambda)y)\in B(\Omega).

For (2), if x,y∈B−1(Θ)x,y\in B^{-1}(\Theta) then B(x),B(y)∈ΘB(x),B(y)\in\Theta, and convexity of Θ\Theta plus the same identity gives B(λx+(1−λ)y)∈ΘB(\lambda x+(1-\lambda)y)\in\Theta.