2.3: I claim that the definition “m is the infimum of S := ∀x:x≤m⇔x≤S” is better: It should make for prettier proofs, and expresses “m eliminates ∀s∈S for purposes of testing ≤”.
Have this marvelously pretty proof of 3b: Theorem: ∀p∈{monotone, 1-Lipschitz, 0-increasing, concave}:pΦ⟹pinfΦ Proof:
ϕ is monotone iff ∀x≤y:ϕx≤ϕy. ϕ is 1-Lipschitz iff ∀x,y:ϕy−|y−x|≤ϕx. ϕ is 0-increasing iff 0≤ϕ0. ϕ is concave iff ∀C convex combination:ΣϕC≤ϕΣC. It thus suffices to show that for enough α,β:αΦ≤βΦ⟹αinfΦ≤βinfΦ.
We’d like to say αinfΦ≤infαΦ≤infβΦ≤βinfΦ. The first needs precisely that α be monotone, which all of the above are. The second is inf being monotone at the premise. The third is equality when β is ϕ↦ϕx (which all of the above are), characterizing the very definition of ≤ on functions. □
Open: Does 3a characterize affine? Can the full set of p for which this goes through be succinctly described?
5c:
I had rejected this answer as unfair. This sort of trolling should be overseen personally.
2.3: I claim that the definition “m is the infimum of S := ∀x:x≤m⇔x≤S” is better: It should make for prettier proofs, and expresses “m eliminates ∀s∈S for purposes of testing ≤”.
Have this marvelously pretty proof of 3b:
Theorem: ∀p∈{monotone, 1-Lipschitz, 0-increasing, concave}:pΦ⟹pinfΦ
Proof:
ϕ is monotone iff ∀x≤y:ϕx≤ϕy. ϕ is 1-Lipschitz iff ∀x,y:ϕy−|y−x|≤ϕx. ϕ is 0-increasing iff 0≤ϕ0. ϕ is concave iff ∀C convex combination:ΣϕC≤ϕΣC. It thus suffices to show that for enough α,β:αΦ≤βΦ⟹αinfΦ≤βinfΦ.
We’d like to say αinfΦ≤infαΦ≤infβΦ≤βinfΦ. The first needs precisely that α be monotone, which all of the above are. The second is inf being monotone at the premise. The third is equality when β is ϕ↦ϕx (which all of the above are), characterizing the very definition of ≤ on functions. □
Open: Does 3a characterize affine? Can the full set of p for which this goes through be succinctly described?
5c:
I had rejected this answer as unfair. This sort of trolling should be overseen personally.