Notes on the differentiation of quasi-convex functions

Lemma 1.2. (D at UCP). Suppose u is quasi-convex. If x is an upper contact point for u, then u is differentiable at x. Moreover, if (p, A) is any upper contact jet for u at x, then p = Dx is unique. Another even more standard result is called partial continuity of the gradient, or first derivative.

Lemma 1.3. (PC of FD). Suppose u is quasi-convex and xj →x. If u is differentiable at each xj and at x, then Dxju →Dxu. The next two results concern the second-order contact of quasi-convex functions and are of a deeper nature. ∗Partially supported by the N.S.F. 1

THEOREM 1.4. (Alexandrov). A locally quasi-convex function is twice differentiable almost everywhere. For the next result we need two variations of the notion of an upper contact jet. First, we say that (p, A) is a strict upper contact jet for u ∈USC(X) at x0 ∈X if the upper contact inequality (1.1) is strict for y ̸= x0. An understanding of the strict upper contact jets will be adequate for our discussion since (p, A) is an upper contact jet if and only if (p, A + ǫI) is a strict upper contact j

THEOREM 1.7. (Jensen-Slodkowski). Suppose that u is a quasi-convex function possessing a strict upper contact jet (p, A) at x. Let Bρ denote the ball of radius ρ about x. Then there exists ¯ρ > 0 such that the measure |C(u, Bρ, A)| > 0 ∀0 < ρ ≤¯ρ. (1.3) This result follows in a straightforward/elementary manner (see Section 4) from Slod- kowski’s Lemma 4.1 below, which in turn is proved in Sections 5–7. On the other hand, Slodkowski’s Lemma 4.1 and Jensen’s Lemma 9.1 below are equiv- alent speci

THEOREM 1.8. (Upper Contact Jets). Suppose u is quasi-convex with an upper contact jet (p0, A0) at a point x. Then (D at UCP) u is differentiable at x and Dxu = p0. Suppose E is a set of full measure in a neighborhood of x. Then there exists a sequence {xj} ⊂E with xj →x such that u is twice differentiable at each xj and (PC of FD) Dxju →Dxu = p0, (PUSC of SD) D2 xju →A ≤A0.

Theorem 1.8 can be stated succinctly in terms of the subset J+(u) ⊂J2(X) ≡X × R×Rn×Sym2(Rn) of upper contact jets for u, and another subset depending on E. Define J(u, E) ⊂J2(X) to be the subset of tuples (x, u(x), Dxu, D2 xu + P) such that x ∈E, u is twice differentiable at x, and P ≥0. Then Theorem 1.8 condenses to: If u is quasi−convex and E has full measure, then J+(u) ⊂J(u, E). (1.5) We will deduce the four results from the special case where u is convex, and for Lemma 1.7 we will reduce to t

Lemma 2.1. |u(y) −u(x)| ≤C|y −x| (2.6) where the Lipschitz constant C is the supremum of |p| taken over p ∈∂u(x), x ∈K. The second is the following.

Lemma 2.2. u is differentable at x ⇐⇒ ∂u(x) = {p} is a singleton, in which case p = Dxu = lim y →x q ∈∂u(y) q. (2.7)

Corollary 2.3. u is differentiable everywhere ⇐⇒ ∂u is single valued ⇐⇒ u is C1.

Lemma 2.5. Suppose ϕ(y) ≡c + ⟨q, y⟩+ 1 2⟨Py, y⟩with P ≥0. Then ∂(u + ϕ)(x) = ∂u(x) + ∂ϕ(x) = ∂u(x) + q + Px.

Lemma 4.1. (Slodkowski). Suppose that u is a convex function with a strict upper contact jet (0, λI) at a point x. Then there exists ¯ρ > 0 such that the measure |C(u, Bρ, λI)| > 0 ∀0 < ρ ≤¯ρ. The following trivial lemma is all that is needed for the reduction. First note that a degree-2 polynomial ϕ(y) satisfies ϕ(y) = ϕ(x) + ⟨Dxϕ, y −x⟩+ 1 2⟨(D2 xϕ)(y −x), y −x⟩ ∀x, y ∈Rn. and D2 xϕ is independent of x.

Lemma 4.2. Suppose ϕ is a degree-2 polynomial. Set B ≡D2 xϕ. (1) If (p, A) is an upper contact jet for u at x on X, then (p + Dxϕ, A + B) is an upper contact jet for w ≡u + ϕ at x on X. (2) (p, A) is strict for u ⇒(p + Dxϕ, A + B) is strict for w ≡u + ϕ. (3) C(u + ϕ, X, A + B) = C(u, X, A)

Lemma 5.1. There is an open vertical slab SLAB ⊂Rn+1 written as the intersection SLAB = H1 ∩H2 of two parallel vertical open half-spaces with the following property. Let ch ≡ch(epi(ϕ1) ∪epi(ϕ2)). Then graph(ϕ1) ∩ch ⊂H1 and graph(ϕ2) ∩ch ⊂H2 (5.1) Moreover, the width of SLAB is |v1 −v2|.

Lemma 6.1. If x ∈C(u, X, 1 r I, v), then |x −v| ≤ p 2rOscX(u).

Lemma 6.1′. Set δ ≡ p 2rOscX(u) and Xδ ≡{y ∈X : dist(y, ∂X) > δ}. For any point v ∈Xδ the contact set C(u, X, 1 r I, v) is a non-empty compact subset of the open set Xδ. In fact, it is contained in the closed ball Bδ(v) about v of radius δ. The Upper Vertex Map Now suppose X ⊂Rn is open and u(y) ≤u(x) + ⟨p, y −x⟩+ 1 2r|y −x|2 ∀y ∈X. (6.5) By Definition 1.5 x ∈C(u, X, 1 r I), that is, x is a global upper contact point of type 1 rI

Proposition 7.1. Given a convex function u defined on an open convex set X ⊂Rn, the vertex map V : C(u, X, 1 r I)) →Rn is a contraction.

Proposition 8.1. If u is a convex function on Bρ satisfying 0 ≤u(y) < |y|2 2R for y ̸= 0. Then for 0 < r < R, |Bρ|  1 − q r R n ≤ C u, Bρ, 1 r I

Lemma 9.1. (Jensen). Suppose that w is a quasi-convex function with the strict upper contact jet (0, 0) at x (equivalently, w has a strict local maximum at x). Then there exists ¯ρ > 0 such that |C(w, Bρ, 0)| > 0 ∀0 < ρ ≤¯ρ. In a very strong sense the Slodkowski Lemma is equivalent to Jensen′s Lemma. (9.1) The precise statement (9.1) is embedded in the following proof.

Lemma 10.1. The multi-valued map ∂f is expansive. That is, if (x1, y1) ∈∂f and (x2, y2) ∈∂f, then |x1 −x2| ≤|y1 −y2|. 13

Lemma 10.2. Xδ ⊂Dom(G) ≡Y where δ = 2 p r|u|∞. (10.5)

Lemma 10.3. The Legendre transform g of f is a convex C1 function on Xδ with derivative Dg = G. If G is differentiable at y with first derivative DyG = B, then g is twice differentiable at y with second derivative D2 yg = B.

Lemma 10.4. Suppose that G is differentiable at y0 ∈Xδ and let B ≡Dy0G denote the derivative. Assume that x0 = G(y0) is not a critical value of G. Further assume that the convex function f is differentiable at x0, and hence Dx0f = y0. Then the function f is twice differentiable at x0 with second derivative D2 x0f = B−1.

Lemma 2.2. Since y0 = 0 is not a critical point of G, the derivative D0G ≡B is invertible. Let A ≡B−1. We must show that: f(x) −1 2⟨Ax, x⟩= o |x|2
. (10.10) For (x, y) ∈∂f the identity f(x) + g(y) = ⟨x, y⟩can be written as f(x) −1 2⟨Ax, x⟩= 1 2⟨By, y⟩−g(y) + 1 2⟨y −Ax, x⟩+ 1 2⟨x −By, y⟩. (10.11)

Lemma 10.4 applies to each point x0 = G(y0) ∈D with y0 /∈N ∪C.

Lemma 10.5. The set D −G(N ∪C) has full measure in G(Xδ).