Definition. A set C⊆Rn is convex if for any points x,y∈C and λ∈[0,1].
λx+(1−λ)y∈C Familiar Set Convex
Line: fix any Z∈Rn. 0=d∈Rn.
L={Z+td∣t∈R} Proof:
λx+(1−λ)y=λ(z+td)+(1−λ)(z+td)=λz+(1−λ)z+λdtx+(1−λ)dty=z+(λtx+(1−λ)ty)d
Hyperplane: Hα,β=x∈Rn∣aTx=β. where a∈Rn\{0}&β∈R
Ex:
He,1={x∣eTx=1}={x∣j=1∑nxj=1}
Halfspace: Hα,β−={x∈Rn∣aTx≤β}(0=a∈Rn,β∈R)
Norm balls: B□={x∈Rn∣∥x−C∥□≤r}, where C∈Rn (center) and r∈R+ is radius
Proof: A norm ∥⋅∥:Rn→R+ satisfies the following
∥αx∥=∥α∥⋅∥x∥,∀α∈R
∥x+y∥≤∥x∥+∥y∥
∥x∥=0⇔x=0
Example: The set of positive semidefinite matrices
Sn×n≡{x∈Rn×n∣x ane PSD} is convex
Z=λx+(1−λ)y, where X,Y∈Sn×n then Z is positive semidefinite.
Operations on sets that preserve convexity:
Intersection
For any collection of convex sets Ci∈Rn, i∈I, then i∈I∩Ci is convex. The union does not preserve convexity.
Ex: the unit simplex in Rn is the set Δn:={x∈Rn∣∑j=1nxj=1,x≥0}
He2,0−={x∈Rn∣x2≥0,x1∈R} Proof: Take x,y∈i∈I∩Ci, show λx+(1−λ)y∈i∈I∩Ci,∀λ∈[0,1].
Because x,y∈i∈I∩Ci⟹x,y∈Ci∀i∈I
Because Ci convex λx+(1−λ)y∈Ci∀i⟹λx+(1−λ)y∈i∈I∩Ci
Addition. If C1,C2,…,Cm are convex sets in Rn, then the set addition
C1+C2+…+Cm={Z=x1+x2+…+xm∣xi∈Ci,i=1,…,m} is convex.
Image of a set. If C≤Rn is convex and A is an m×n matrix, then A(c):={Ax∣x∈C} is convex.