Mazur [10] proved that, under the assumption that X is separable, the set of all smooth points of Cis a dense subgroup G6 of the limit of C. The former consists of those Banach spaces in which any continuous, convex function defined on an open, convex set is Gateaux differentiable on a dense subgroup G, of its domain. The Lipschitz function defined on an almost open convex set is Gateaux differentiable on a dense G.s subgroup of its domain (see [23]).
The set qi(A) of all quasi-interiors of A is called the q1wsi-interior of A. Since Kx(A) = Kx(cl (A)), it follows that any dense subset of a quasi-open set is quasi-open. Clearly, the interior of a convex set is contained in its quasi-interior.
As we will see, it can happen that the interior is empty, while the quasi-interior is not empty. However, if the interior is not empty, it is equivalent to the quasi-interior. Since int(C) has a density inC, the previous inequality holds for every y EC and thus for x E S( C). 1) It is well known that if X is finite dimensional, then a convex subset C of X has a non-empty interior if and only if the affine hull of C is X.
We have already mentioned (Remark 2.7 (2)) that if Cis a closed convex subgroup of the Banach space X with empty interior but non-empty quasi-interior, then both qi (C) and S (C) are dense in C. Let C be a closed convex subgroup of the Banach space X. of a topological space T is called F,. respect a G6) if it is the union (respectively intersection) of many countable closed (respectively open) subsets of T.
Chapter II
MINIMAL USCO MAPS
To prove that G is upper semicontinuous at a, let U be an open subset of Z such that G(a) ~ U. Then there exists an open neighborhood U of z and an open set W such that F (a) contains such that U n TV= 0. The fact that H is usco implies that there exists an open neighborhood U of such that H(U) ~ lV1.
Since His is upper semi-continuous, there exists an open neighborhood U of a0 such that H(a) CH!- for all a E U. An interesting connection exists between the fact that a minimal usco map F is single valued in a point and the continuity of a selection for F at that time. 22 such that Fo ::::S F; since any choice of F0 is also a choice of F, the claim follows from the definition of the class C and Lemma 4.1.
Assume that there exists a metric d on Z with the following property: every open nonempty subset U of B bas a.n open nonempty subset V such that F(V) contains nonempty, relatively r-open subsets of arbitrary small d diameter. Since F is minimal, by Proposition 3.6 there exists a nonempty open subset G contained in V such that F(G) ~TV, hence diam(F(G). Ba.nacl1 space such that X* has Ra. clon -Nikodjrm property and let F : A -+ 2x• be minimal (convex) with usco map where A is a Baire space.
We cannot use the lemma that follows from this corollary to find a dense, relatively open subset D of A such that F I D: D -+ 2-'<* is a locally bounded, minimal w* usco mapping. By Proposition 4.4, there exists a dense G 6 subset Ao of A such that F( x) has one value and norm usco for every x E Ao. Then there exists an open, dense subset D of A such that FID: D ---t 2x• is a locally (norm) bounded, minimal (convex) w* usco map.
Then there exists a dense G0 subset Ao of A such that F( x) is contained in the sphere of X* for eveq x E Ao. Since II II w* is lower semicontinuous, we can find w* a point of the cluster x* E F(a) of the lattice (x~) such that the first lower inequality is satisfied. As above we can find the w* cluster point x* E F(a) of this lattice such that liminf llx~ll 2: llx*ll· Using the continuity of 7/J at a we get.
MONOTONE OPERATORS
Locally Efficient Monotone Operators
She x E B(a,t:) n A andy EX; since x E qi(A) there exists a sequence (tn) of positive real numbers that converge to 0 and a sequence (Yn) C X that converges to y such that x + tnYn EA. We will see shortly (Corollary 6.3) that locally efficient maximal monotonic operators defined on quasi-open subsets are minimally convex w* usco maps and therefore they inherit many of the properties of the minimally convex w* usco maps. However, some of these properties hold for maximal monotonic operators defined on not necessarily quasi-open sets, where the proofs are more or less similar.
Let A be a subset of the Ba.na.ch space X and let T: A --+ 2x• be a locally efficient maximal monotonic operator. A is also normal, there still exists an open lid (Wi)iEI of A such that clA(Wi). It is easy to see that~;; is continuous on A. It is clear that Tt/J is a locally bounded, monotonic operator.
2x· is a locally efficient maximal monotone operator, then there exists a dense G6 subset D of A sud1 such that T0 is w* usco at every point of D. Tl1en there exists a dense, relatively open subset D of A such that T is efficient at every point in D. The following result shows that monotone operators on certain domains have unique maximal monotone extensions (over the same domain).
If A is quasi-open on X, then M can be described as follows: let T be a monotonic operator on A whose graph is the closure (in A x X*) of the graph T; then A1 ( x) w* is a closed convex hull T( x. To prove that M is unique, suppose that there exists another maximal monotone operator S on A containing T. From Proposition 5.10, there exists a dense, relatively open subset D1 from A such that T is efficient at every point from D1.
Then there exists a dense G0 subset D of A such that T(x) is contained in the sphere X* for every x E D. This follows directly from the previous theorem and the fact that the monotone operator is always contained in the maximal one. Let A be a quasi-open Baire subset of the Banach space X and let T: A---> 2x• be a monotone operator.
Then there exists a dense G0 subset A0 of A such that T is single-valued and norm-to-norm (resp. norm-to-w*) upper semi-continuous at every point of Ao. Since A is quasi-open, Theorem 6.6 (1) implies that there exists a dense, relatively open subset D of A such that TID is a locally bounded, minimal convex w* usco map.
Chapter IV
SUBDIFFERENTIABILITY AND DIFFERENTIABILITY
Convex Functions
Then C0, if nonempty, is a convex subset of X and the pointwise. supremum supfi is a convex function on Co. 5) Assume that C is an open convex subset of X and that f: C ---t R is continuously differentiable and has a second derivative f" through C. Namely, it is sufficient for them to be bounded from above becomes an open subset of their domain. It remains to show that f is bounded from above in the neighborhood of every point of D.
On the other hand, it is not difficult to verify that f is not locally Lipschitzian at any x in C (according to the previous proposition, this shows that the interior of C is empty). We will call f (X) :mbdiferencia.l of f at X. 2) We call the function f sublidifferentiable at ;z; if its subdifferential at x is not empty, i.e. there is at least one subgrad.dient outside of a.t x. On the other hand, using first the definition of .f~ and then the convexity of f, we have
I have seen (Lemma 8.2) that a necessary condition for the subdifferentiability of f a.t x is that f is lower semi-continuous at x. The following example (due to I3orwein and Fitzpatrick) will show that even if A is a dense G0 subset of C, the theorem does not hold. Under the same assumptions as in the previous theorem, the restriction of a.f to A is a maximal monotone operator on A.
Let C be a quasi-open convex subgroup of X, and A be a relatively open Baire subset of C (eg, A= Cis quasi-inside a closed convex subgroup of X). If its domain is open, the answer is yes: being subcustomizable, f is semicontinuous (see Lemma 8.2), so it is locally Lipschitz (see Proposition 7.2; in fact we only need the subcustomability outside in one point!). If in addition .f is lower semicontinuous in clc(A), the closure of A inC, then .flclc(A.) is locally Lipschitz at every x EA.
Then there exists a dense, relatively open subset D of A such that the constraint on A is locally Lipschitz at every point of D. If, moreover, .f is lower semicontinuous on clc(A), then the constraint on clc(A) is locally Lipschitz on at every point D. Since .f is n-efficient on An, it is locally efficient on D and the theorem follows from the previous statement.
Gateaux and Frckhet Differentiability
Conversely, if x E qi(C), .f is Lipschitz in a neighborhood of x and 8.f(x) is a single tone, then f is Gateaux differentiable at x. Let f: C --t R be convex and locally Lipschitz on a relatively open subgroup A of the interior of C. If a is continuous at x, this immediately shows that f is Frechet differentiable at x (and that a( x) is a Frechet differential of f a.t :r).
From the previous lemma, C0 consists of points C in which a is continuous from norm to norm, which is. known that G0 is a subset of C. Asplund spaces can alternatively be defined as those Banach spaces with the property that every convex and continuous function defined on an open convex subset X is Frechet differentiable at at least one point of its domain. Indeed, if a Banach space X satisfies this last condition, then any continuous convex function f : C --+ R, where C is open and convex, is Frechet differentiable on a dense subset Co in C and consequently 0.6 also Co Go a subset of C .
Conversely, let X have the stated property and let f: C - t R be a continuous convex function, with C open. Then f is a minimal map w* usco, hence, by Corollary 4.7, it is norm continuous at every point of a dense subgroup G6 of A. By assumption there exists a dense subgroup G6 Co of C1 such that <7 is norm continuous at every point of C0.
Then there exists a dense G6 subset Ao of A such that .f Gateaux is differentiable at every point of A0. Then the set of points of C where f is not Gateaux differentiable can contain a Ba.ire, quasi-open convex subset of C. Let h: B--+ A be continuous and Frechet differentiable (resp. Gateaux differentiable) and f: C --+ R be convex and such that f I A.
Then there exists a dense subgroup G6 B0 of B such that f o h is Frechet (respectively Ga.tea.ux) differentiable at every point of B0. By Corollary 4.7 there exists a selection a for F which is norm continuous at every point of a dense subgroup G0 B0 of B. In our case C is arbitrary and imposing stronger conditions on f we obtain its subdifferentiation at every relatively open subgroup of C. in which it is locally Lipschitz.
