ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I. PRACE MATEMATYCZNE XXV (1985)
Wie sl a w Kurc (Poznan)
On some dichotomy concerning strict convexity for integral functionals
1. Introduction. The main intention of this paper is to explain an aspect of strict convexity for a class of integral functionals. Let us first fix notations and definitions that will be used throughout the paper.
Let У be a real linear space and let F : Y - * R v { + 00} denote a functional from C onv(7); i.e., dom (F) = {уеУ: F(y) < 00} ф 0 and F is convex. It is clear that F is convex if and only if epi(F) = {(у, r)e У x x R : F{y) ^ r] is a convex set. F is said to be strictly convex if F is finite on У and the following equality F(Ay + ( l —A)z) = AF(y) + ( \ —A)F(z) with Ae(0, 1) implies y = z. In the following definition an extended concept of strict convexity is given.
De f in it io n 1.1. Let & be a family of subsets of У. A functional F will be called J *-strictly convex if the projection of any face of epi (F) into У belongs to some set from
The faces of a convex set К in a linear space are exactly the sets К n M , where M is an affine subspace supporting К in the sense that K n M Ф 0 and K \ M is convex.
The property of ^ -s tric t convexity may be localized to any convex subset K cz Y. A functional F e Conv (У) is said to be -strictly convex on К if the functional F + 0K is ^-strictly convex. For the sake of clarity in this case it will be always assumed that К c= dom(F). It is obvious that if dom(F) с K, then ^ -s tric t convexity of F coincides with «^-strict convexity of F relative to K.
Ex a m p l e s. Let & — 2Y. Then no restriction is imposed on F. Hence any F eConv (У) is 2r -strictly convex. The opposite case arises when
= {{у}: y e У}. In this case it is intuitively clear that ^ -stric t convexity of F e C o n v (y ) coincides with the usual strict convexity of F on dom(F).
In this paper we deal mainly with -strict convexity for integral functionals with = {A a Y: dim(zl) < k}, where 1 < к ^ 00; dim(A)
= dim(aff(zl)). When this is the case F will be also called k-strictly convex relative to dom(F) or simply k-strictly convex when dom(F) = У We remark
that -strict convexity implies J^'-strict convexity of a functional F whene
ver k < / .
For given u e Y let 3Fk = u {aff({u} и A): A e ^ k}."It is obvious that -strict convexity implies J^-strict convexity for all 1 ^ к ^ oo, и e Y As a typical example let us consider a semi-norm F ( - ) = ||-||y, It is easy to see that F is not -strictly convex. However, F is 'Strictly convex exactly when the semi-normed space Y is strictly convex in the usual sense. Also it may be easily checked that .^ 'Strict convexity, where 1 ^ к < oo, coincides with the well-known notion of к-strict convexity introduced by I. Singer, cf [9], when F denotes a norm on Y It follows in particular that the notion of
^ ‘-strict convexity is more adequate than the notion of -strict convexity for a functional F e C o n v (Y) when the later is homogenous with regard to the point u e Y
In the following propositions some equivalent conditions for k-strict convexity are given.
Proposition 1.1. The following statements on a functional F eConv(Y) are equivalent (when considering (c) assume 1 ^ к < oo):
(a) F is ^ k-strictly convex.
(b) Every face o f epi (F) that is non-vertical is either finite-dimensional when к = oo, or o f the dimension less than к otherwise, i.e., when 1 ^ к < oo.
к к
(c) Let 1 ^ к < oo. For y 0, ykedom (F) i f F ( £ Я,у() = £ AfF (yf)/o r
о о
к
some Я,- > 0, i = 0, .. ., к, £ Я,- = 1, then y0, ... , yk are affinely dependent.
о
Let us mention that a face of epi F is non-vertical if it lies on a hyperplane К — {(у, r)e Y x R : y*(y) + sr = d] with s Ф 0.
Proposition 1.2. Let l ^ k c o o . The following statements on F e C on v ( Y) are equivalent:
(a') F is 'jFo-strictly convex.
к к
(b') For y0, ..., ykgdom(F) if F ( £ 2,yf) = £ far some Я,- > 0,
о 0
к
i = 0, . .. , k, Yj Я; = 1, then y 0, . . ., yk are linearly dependent.
о
These propositions were inspired by the results of I. Singer, cf. [9], concerning characterization of к-strict convexity of a norm.
In the following we give a definition of the class of functionals that will be a basis for our considerations. We start with a brief summary of assumptions usually imposed on integrands in the defining of the integral functionals. Let {Q, Г, fi) be a measure space, where I denotes a or-algebra of subsets of a given set Q and p denotes a positive tr-finite and complete
measure on I . The letter X always denotes a separable Banach space over the reals.
In accordance with [2] we say that a function /: Q x X R u { + oo}is a normal convex integrand if:
(i) for \i-a.a. coeQ the functional x-+ f(co, x) is l.s.c.,
(ii) / is X x ^-m easurable function, where ^ denotes a a-algebra of Borel subsets of X,
(iii) f(co, -)eConv(Y ) for g-a.a. coeQ.
If, moreover,
(iv) / (со, •) ^ 0 on X, f (со, 0) = 0 ( / (со, — x) = / (со, x) for all x eX ) yU-a.e. on Q we shall say that / is Young's integrand (resp. even Young's integrand).
Finally / will be called a Young's function if it is even Young’s integrand and satisfies a coercivity condition:
(v) for each r ^ O the following sections f ^ r(co, •) = [ x e X : f (со, x) ^ r] are norm-bounded in X.
It may be easy verifed that the following statements are equivalent to the statement (v):
(va) ||x„|| -» + oo implies that / (со, x n) -* + oo /r-а.е. on Q,
(vb) lim f(co, r) = +oo where f(co, r) = inf {f (со, x): ||x|| = r), i.e.,
r —> + 00 _
/(со, •) is uniformly large near infinity for /r-а.а. соe ü ; cf. [12],
(vc) there exist functions a, / : Q -> R+ \ {0} such that ||x|| ^ /3(co) implies /(со, x) ^ а (со) /г-а.е. on Q (in fact the function a may be taken as a
constant function on Q).
Moreover, condition (v) implies the following two conditions that are equivalent each other:
(v'a) there exists a function ft: Q -> R + \{0} such that ||x|| ^ f(co) implies /(со, x) > 0 for /т-а.е. coeQ,
(v'b) for every О Ф x e X, f (со, rx) -> + oo if r -> + oo.
All these (v)-conditions coincide when Y is a reflexive Banach space.
Given a Young’s integrand / : Û x X -> R+ u {+ oo}, we define
(1.1) Ff (u) = f/(ft), u(co))dg,
o
where u eMx(Q) and M X(Q) denotes the space of all (strongly) measurable functions u: Q- +X. In everything that follows M X(Q) is assumed to be infinite dimensional. It is clear that Ff is the convex functional on M X(Q) with property that Oedom^Fy). In other words we have Ff e C o n \ ( M x). The most pathological situation arises when / = 0 on ü x X. Then, the face of epi Ff reduces to the whole hyperplane M x ( Q ) x { 0} in M x ( Q ) x R 1. In this case Ff cannot be ^ -s tric tly convex with any 1 ^ к ^ oo.
Our aim is to prove that the absence of ^ - s t r i c t convexity for
1 ^ к^ oo is of frequent occurrence for Ff in the sense that for p continuous (free of atoms) Ff is either not J^-strictly convex for every l ^ f c ^ o o o r F y is already -strictly convex. Moreover, we shall prove that, for continuous measures p, Ff is -stictly convex precisely when the integrand f defining (U ), is ^ 1 -strictly convex.
It will be often used the following abbreviation: “/ has the property W (on a set А с X)” when f(co, •) has the property W (resp. on the set A) for
^-a.a. coeQ.
2. Auxiliary results. First we shall prove the following simple but useful lemma.
Lemma 2.1. Let g: X -* R + u { + oo} be a function such that g(0) = 0, g e Conv (X ) and int (dom (g)) Ф 0 , i.e., g is continuous at zero. I f for some x 0e X and XeR+ u { + oo} g( x0) ^ Я (g(x0) ^ Я; x0 Ф 0 or Я > 0), then for every a > 1 (resp. 0 < a < 1) there exists a neighbourhood U o f the point ax0 such that g ( U ) ^ Я (resp. g(U) ^ Я).
P ro o f. The case of Я = 0 is evident. Let g( x 0) ^ Я > 0. Assume conver
sely that there exists a > 1 such that for every neighbourhood U of ax0 there is x e U for which g (x) < Я. The convexity of g yields
Taking x - > a x 0 and making use of the continuity of g at zero we get a contradiction. The proof of the second case, when g (x0) ^ Я, is quite similar to the given above. It suffice to consider the easily verified inequality:
Now, let /: Q x X R+ и { + oo} be a Young’s integrand. Define the following relations in ü x X :
(21) Г° = К0** *)' *> > y i(")}’ Г = {(m, x): f(co, х ) >Ц( о) } , Л0 = {(со, x): f(co, x) < Я(ю)}, A = {(со, x): / (со, x) ^ Я(о>)},
where Я: Q R + u { + oo} is a given measurable function. We shall refer to the terminology of [3]. In particular, when considering a relation Г c: Q x X we shall refer to Г rather as to the function a> -*■ Г (to) from Q to 2*, where F(œ) denotes the section of Г in со. Thus F will denote both the relation and the function. If со-* Г (со) is a function from Q to I х let G r (Г)
= {(со, x )e Q x X : хеГ(со)} and d o m (r) = Г ~ 1 (X). Thus, given a relation Г, then Gr(T) and dom (Г) corresponds to the associated function со -* Г (со) ; plainly in this case G r (Г) — Г. To each relation or function corresponds a
subset of Q. Following [3], when dom(F) = Q a relation (or a function) Г is called a multifunction.
The following proposition completes for our purposes the well-known Theorems 6.2-6.4 from [3]. Our aim is to prove a weak measurability of frequently used relations (2.1) without assumption on the continuity of the integrand / on X except the point zero, where / is assumed to be continuous.
Pr o p o s it io n 2.1. Let X be a separable Banach space and f is continuous at zero.
(a) All the relations above defined are weakly measurable. Moreover, the relations Г 0 and A are measurable with open and closed values respectively;
i.e., the sections Г 0(со) (A(co)) are open (and closed) in X , respectively.
(b) dom (F0), dom(F), dom (/t0), dom (/l)eZ ’; dom(yl) = Q, dom(A0) = Q when X > 0.
(c) The graphs o f all above defined relations are I x 3$-measurable. Thus A and A 0 for X > 0 are measurable multifunctions.
(d) The integrand f is necessarily I x measurable, i.e., condition (ii) from Section 1 may be given in a weaker form: <o->/(co, x) is measurable function for all x e X and f is continuous at zero.
P ro o f. The method of proving is classical, therefore the proof will be sketched only. The measurability of the relation T 0 follows on account of I x ^-measurability and l.s.c. of /. The proof of this fact is similar to the proof of Theorem 6.2 in [3]. Condition (i) implies that the sections T 0(a>) are open in X.
Now, let 0 be any open set in X . Let A be a countable dense subset of X. We shall prove that
(2.2) Г- 1(0) = U / М x) > А(ш)} = (J {o: f(to , a) > A M ].
xeO a e O n A
Let с о е Г ~ 1(&). Then /(со, x 0) ^ X(co) for some x 0e X . Hence, by virtue of Lemma 2.1, for each a > 1 there is a neighbourhood U of the point ax0 such that f(co, U ) ^ X(со). Choose a > 1 such a way that ax0e&. Then V
= (9 n U is a neighbourhood of ax0 and hence А с л У Ф 0 , i.e., / (со, a)
^ X(co) for some a e O n A . This proves the first inclusion. The converse inclusion is clear and hence (2.2) follows.
Assumption (ii) for / and representation (2.2) implies that T ~ 1( 0 ) e l , i.e., the weak measurability of Г.
The proof for A follows by virtue of Proposition 2.2 and Theorem 3.5 from [3] because A = Gr(A) is I x ^-m easurable and A(ofi is closed by l.s.c.
of f(co, •) for /i-а.а. coeQ.
To get the proof for A 0 let 0 cr X be an open set. Let cogAq1 (0). Then if / (со, x) < X(ofi, then the same holds for a e O n A , by virtue of lemma 2.1.
Therefore, A ô 1^ ) = Л 0 х((9п A) e Z , even in the case when A ô 1 (C) = 0 .
6 — Roczniki PTM — Prace Matematyczne XXV
Thus Л о is weakly measurable and hence d o m ^ o J e Z 1. Moreover, d o m /4 0)
= Q when X(co) > 0 on Q.
Finally, to prove (d), let X be any (finite) real number. It suffices to prove that A — {(со, x): /(со, x) ^ A} ( = вг(Л )) e l x j knowing only that instead of (ii) / (со, •) is continuous at zero and со - ^ f (со, x) is measurable for all x e X (we assume, however, that / satisfies conditions (i), (iii), (iv) from Section 1).
Let (9 с X be any open set. Then, by virtue of Lemma 2.1, A - 1 {(9) = IJ (со: /(со, x) ^ X]
хвВ
= U {m: /(со, a) ^ (со: /(со, 0) ^ l } e l x J .
a e O n A
Therefore, the function со ^ A (со) is weakly measurable and hence, by Theorem 3.5 from [3], has I x J ’-measurable graph A, because A (со) is closed by virtue of l.s.c. of /(со, •) for /с-а.а. соей.
Let us note that thesis (c) is an immediate consequence of condition (ii) for / from Section 1 and of measurability of the function X(co).
3. Main results. Let /: Q x X -*■ R+ u {-f oo} be a Young’s integrand.
Following [2] let Cf ~ dom(Fy), where Ff is defined by (1.1). Under these assumptions on / the set Cf is non-empty and convex. In the following theorem a necessary and sufficient condition for -strict convexity of Ff is given.
Th e o r e m 3.1. (a) I f f is # ' 1-strictly convex, then Ff is ^ - s t r i c tl y convex.
(b) I f [i is a continuous measure and f is continuous at zero, the converse is also true.
R e m a rk . In the second part of the theorem, we have Cf Ф {0} because / is continuous at zero, however, in the first one it may happen that Cs = |0}.
Co r o l l a r y 3.1. Let ц be continuous and let f be continuous at zero. Then Ff is strictly convex relative to Cf if and only if f (co, -) is strictly convex relative to dom (/(co, •)) for ц-а.а. со е й .
Let us note that the results closely related to those in the above corollary, in a connection with a problem of strict convexity of Minkowski norm generated by means of Fy-type modulars, have been obtained by many authors for special kinds of Ff . In fact we were inspired by papers [1], [4], [11] in proving of this corollary (cf. also [5], [7], [6], [10]).
P r o o f o f T h e o r e m 3.1. (a) Let / be ^ - s tr ic tly convex. If Cf = {0}, there is nothing to prove; all projections of the faces of epi(Fy) into M X(Q) reduce to zero-dimensional set {0}. Hence we will assume that Cf Ф (0). Let u, v e C f and и Ф v. Let us denote A = {со е й : u(co) Ф v(co)}. We have ц(А) > 0 and f(co, u(co)), f (со, v(co)) < +oo for all coeQ except a set of measure zero.
On account of J* 1 -strict convexity of / and Proposition 1.1 (a), (c) we obtain
/(со, Лм(со)+(1 — A)r(co)) < Д/(со, w(co)) + (l — A)/(co, t/co))
for 2e(0, 1) and coeA, because u{co), r>(co)edom(/(co, •)). Hence Ff (Àu + + ( l —À) v) <AFf (u) + ( l —À)Ff (v), Ле(0, 1). This proves J^-strict convexity for Ff if one applies again the equivalence of (a) and (b) in Proposition 1.1.
(b) This thesis will follow from our next Theorem 3.2 as a special case, however, an immediate proof is possible (cf. [1], [4], [5]) (a)=>(b).
Thus, it remains to prove that the continuity of / at zero implies С/ Ф {0} (cf. remark above). We argue along the scheme from [2], p. 11. Let us consider the relation
F — {(<o, x)e Ü x X : f (со, x) < a (со)} n Q x(2f\{0}),
where aceLl (Q). The continuity of / at zero implies dom (F) = Q. Moreover, Г is Г x ^-measurable. Hence Aumann’s selection theorem, [3], admits a measurable selector 0 # 0(со)еГ(со), co eQ. It is now clear that Cf Ф {0}.
Remark, in fact we need only a radial continuity of / at zero, but this is equivalent to the continuity of / at zero, cf. [2], p. 13.
The next theorem is the main result in this paper. Roughly speaking, this result with aid of Theorem 3.1 expresses the following dichotomy: either / is -strictly convex (in particular, / is strictly convex on X ) then Ff is S ' 1 -strictly convex or, if / is not strictly convex on X (in particular, / is not J*1-strictly convex), Ff is not ^ -s tric tly convex for every 1 ^ к ^ oo, i.e., epi(F/ ) contains already an infinite dimensional face.
Th e o r e m 3.2. Let ц be a continuous measure and let f: Q x X -► /?+ u
u { + oo} be a Young's integrand that is continuous at zero. Then Ff is J^ 1- strictly convex if and only if Ff is SF*'-strictly convex; precisely, with aid of
Theorem 3.1:
(al) I f f does not satisfy f < +oo, then there exists an infinitely dimensional convex set in M X(Q) contained in the level set F f 1 ( + cc).
(a2) I f f does not satisfy f > 0 on X \ { 0 } , then there exists an infinitely dimensional convex subset o f M X(Q) contained in F f 1 (0).
(a3) Let 0 < / ^ +oo on X \ { 0 } . I f f is not ^ rl-strictly convex, then for every sufficiently small r > 0 there exists an infinitely dimensional convex subset o f Cf contained in the level set F f 1 (r).
R e m a rk . Let us note that something more can be deduced from the proof of this theorem given below. Namely in case (al): for each r > 0 there exists an infinitely dimensional convex subset К c M X(Q) such that Ff (<xK)
< r (= +oo) for 0 ^ a ^ 1 (resp. a > 1). Also, in case (a2): if additionally /(с о ,-) is radialy continuous for /r-а.а. coeO and, moreover,
{coeQ: 3 / (со, x) > 0J = Q (cf. (v'a) in Section 1), then for every r ^ 0 there
exists an infinitely dimensional convex set in Cf contained in the level set F J l (r). In case (a3) r may be any positive real number when / < + 0 0.
P ro o f. The proof will be given in a few steps. First, let us note that by virtue of the definition of ^ - s t r i c t convexity Ff is ^®-strictly convex whenever it is -strictly convex. Therefore, by virtue of Theorem 3.1 it remains only to prove that the lackness of the strict convexity for f on X, implies that some level sets for Ff contain an infinite dimensional convex set, i.e., Ff is not . ^ “ -strictly convex.
P r o o f o f p a r t (al). Our approach here is an adaptation of the method applied in [5] (implication (b) => (a)). Therefore, the proof of this step will be outlined only, especially in its second part.
Let us consider a relation F = {(со, x ) e Q x X: f ( c o , x ) = + o0 }. By virtue of Proposition 2.1, Г is weakly measurable relation, with measurable graph, and therefore dom (Г) = Г~*(Х) is a measurable set. Let us note that /z(dom (Г)) > 0. Hence, by Aumann’s selector theorem, [3], there exists a measurable selection и(ю )еГ(й), со e dom (Г).
Let Г2 = {(со, A ) e Q x R : / (со, Àv(coj) = + 00}. The reasoning based on the same arguments as for Г gives the weak measurability of the relation Г 2 and its graph. Moreover, the relation I / = Г 2 n Q x (0, + 0 0) is also weakly measurable. Now, let A(co) = inf {Л: (со, Я )еГ 1}, сое dom (F 1). By virtue of Theorem 6.6 from [3], Я is a measurable function. Let us denote и (со)
= A(co)v(co), сое dom (Г). Clearly, /(со, au (со)) < + о о ( = + o o ) i f 0 ^ a < l (resp. if а > 1) on dom (Г). Without loss of generality we assume that и is defined on the whole Q, being zero out of dom (Г) (if necessary).
Next, proceeding as in [5] (implication (b) => (a)) and making use of the continuity of p we construct a family {£,■};= ! of disjoint measurable subsets of dom (Г) with finite and positive measures, such that Ff ( at uXs) < l/2 i + 1, i = 1, 2, ..., where а,е(0 , 1) and а, ^ 1 when 1 -> оо. Now we define
00
(3.1) up = X P = l , 2 , . . .
i — P
Thus we have obtained the infinite family of linearly independent functions such that aupe C f when 0 < a < 1 and aup4 C f when a > 1. Moreover, for all positive integers px < p2 -- - < Pn and non-negative reals Ax, X2, ..., An such
П
that Yj h — 1 can be proved that
1
П
a Y k u Pi$ Cf \ а > 1.
i = 1
Let us fix any а > 1. We have proved that the set К = а со({мр}®) is a convex set in M X(Q) of infinite dimension that is not contained in Cf .
Moreover, taking = co([wp) *) with qf sufficiently large the first part of the remark follows.
P r o o f o f p a r t (a2). Let us introduce the relation Г = { (ш ,х )еО х x l : f (cu, x) = 0}. On account of the continuity of / at zero and Proposition 2.1 we obtain that Г' is a weakly measurable relation. Hence, the same is true for the relation Г = Г' n Q x ( X \ {0}). Thus dom(T) is a measurable subset in Q. The relation Г has also I x ^-m easurable graph.
Hence, Aumann’s selection theorem admits a measurable selection ц(ю)еГ(ю), со 6 dom (Г). Clearly, ц (dom(Г)) > 0. Let S c d om (r) be a measurable subset of positive measure and with an arbitrary small measure if necessary. Let 5 => з S2 =>... be a strictly decreasing sequence of subsets in the sense that ju(Sf \ S i +1) > 0, / = 1 , 2 , . . . Define
(3.2) up = uxp, p = 1 , 2 , . . . ,
where и is assumed to be zero out of dom (Г) (cf. Theorem 8.1, [3]). These functions are linearly independent and belong to Cf . Moreover, for all positive integers Pi < p2 < ••• < P„ and non-negative reals Al5 À2, . , A„ such
Л
that £ А,- = 1 : l
od
«= i î i=iw «p,) = o-Therefore the set К = co({up}®) c: Cf and K ci F J 1 (0). This completes the proof of part (a2).
It is possible to obtain something more if one assumes that {со: 3 / (со, x) > 0} = Q and the map £ ->/(co, £x) is continuous as a numeri-
X
cal function (from R to R + и { + oo}), for each x e X and all m eQ except a set of zero measure. In order to get this let us fix an arbitrary r\ > 0, and let us introduce a relation Г = {(со, x ) e Q x X: f (со, x) = rj/p(A)} n A x X , where A cz Q \ S is a measurable subset of finite and positive measure. This is available because S may be of an arbitrary small measure. Then, Г has I x
x ^-m easurable graph with measurable dom(T) = A. Hence, by virtue of Aumann’s selection theorem there is a measurable selector v in Г. We then extend v to whole Q taking у to be zero out the set A. We have now
Ff {K + v) = {/(<», v{(o))dp = rj
A
since Ff (K) = 0. Thus for every r ^ 0: there exists an infinitely dimensional convex set in Cf that is contained in F J x(r).
P r o o f o f p a r t (a3). Let 0 < / < +oo on A^JO}. Let us consider the following relation Г = + ao) n Q x ( X x X \ A ) , where A denotes a
diagonal in X x X and let (cf. [1])
ф(ю, x, у) = / ( w , i ( x + y ) ) - i f { c o , x ) - $ f { w , y).
The set d o m (r) = [ o e Q : 3 ф(со, x, y) ^ 0} is measurable. Indeed, our
X Фу
assumption that / < +oo implies the continuity of / and hence of ф{со, -, •) for /z-а.е. со e Q. Thus a standard argumentation (Theorems 4.5, 6.4, [3]) gives that Г and hence dom(T) are measurable. Moreover, Г is also a E x im
measurable relation. Hence, again by virtue of Aumann’s selection theorem there exists a measurable selector s ={ ul , vt) in Г. Assume s to be extended by zero out dom (Г) to the whole Q. Clearly, u1, v1 are measurable functions and щ (a>) Ф vx (со) for jU-а.е. coedom (r).
Let A c= dom (Г) be a measurable subset of finite and positive measure.
Let us define
A„ = {cueA: f(co, (w )),/(w , ^(m )) ^ и}, ao
where n = 1 , 2 , . . . We have A = (J A„ and, moreover, ... с A„ c A„+l c=...
l
... a A. There exists an integer m such that we have ц (А т) > 0 .
Now it is easy to prove that there exists a measurable set 5 c A m of positive and arbitrarily small measure if necessary, such that on this set:
w(m) = y0WiM + ( l - ' У о К М Ф 0
for some yo G(0, !)• Indeed, otherwise we obtain Ui = on a set of positive measure contained in dom(T), i.e., a contradiction.
Collecting these facts we get that the functions и = ul Xs> v — vi Xs are different each other on S. Moreover, the function w = y0u + (l — y0) ü is different from zero on S and u, v, w e C f since Ff (u), Ff (v), Ff (w) ^ m/i(S)
< + oo. We remark that the numbers Ff (u), Ff (v) are not equal to zero simultaneously. Moreover, one may define functions u2, v2 instead of ul9 by means the formulas: u2 = ux, v2 = when both are different from zero or both are equal to zero, u2 =%vu vi —j vi when ut = 0 and Ф 0, u2
= I ux and v2 = jMi when мх Ф 0 but vx = 0. Defining then и and v as above we obtain, moreover, that Ff (u), Ff (v) > 0. In both these cases for a e [ 0 , 1]:
Ff (au+ {1 —a)v) = aFf (u)+{\ —a)Ff (v).
Now, we are going to the central part of the proof. Let rj > 0 be any fixed real number. Taking, if necessary, a sufficiently small S cz A m one may assume that
0 < Ff (w) = ri0 < ц and n ( Am\S) > 0.
Further, proceeding similarly as at the end of the proof of part (a2) of this theorem we construct a function z e C f in such a way that Ff (z) = rj — r]0
and su p p (z)n S = 0 . To this end let Г = {(со, x ) e Q x X: f (œ, x)
= (*1-П о)/и(Т)} n ( T x X ) , where T a A m\S is a measurable subset of posi
tive and finite measure. Recall that 0 < / < +oo on X \{0}. Thus dom(F)
= T and therefore is measurable. Also, the relation Г has X x J ’-measurable graph. Hence Aumann’s selection theorem admits a measurable selector z in Г. We may assume z to be zero out of T = dom (Г), on Ü. Thus we have supp(w) = S, supp(z) = T and S n T = 0 . Moreover,
(3.3) Ff (w + z) = Ff (w) + Ff (z) = rj.
In particular, we have w0 = w + z e C f .
Now we are able to construct a family of linearly independent functions wl5 w2, ..., contained in Cf and satisfying (3.3), by means of just constructed functions u, t>, w.
Let us consider the following set-functions X n S э B ^ F f (uxB), Ff (vxB).
These functions define continuous (free of atoms) measures of non-negative and finite values. This is a consequence of our construction of the functions u, v. Hence, the following set-function v(B) = (Fj(mxb), Ffivxs)), v: X n S -> R2 defines a vector measure that is also continuous. Now, by Liapunov theorem, there exist measurable sets ... a Sn a S„+1 c ... c cz S with strict (in measure sense) inclusions such that v(5f) = v(iS), i = 1, 2, ...;
Я, е(0, 1), Я, |0 . In particular, this leads to the following equality (3.4) F / (auxsi + (l - a) % .) = Ff (ctu + {1 -ct)v),
where i = 1, 2, ... and a e [ 0 , 1]. Further, let us choose real numbers a(,
& е(0, 1) in such a way that we have агЯД1 — y0) = (1 — Я^&Уо, i = 1, 2, ..., where y0 is the same as in the definition of the function w given above. Now define m, = a, u-t-(l — a,)w and v( = jS, i;-t-(l —jS,)w. It is not hard to see that (3.5) w = Я* м,+(1 — Я,-) г,- ; f = l , 2 , —
Indeed,
Я( иг + (1 — Я£) i7f = A{ <XiU + Ai(l — (Xi) w + (l - AJ v + (1 - AJil - fii) w
= (а,Я;/у0 + Я1( 1 - а 1) + (1 -Я 1.)(1-^))>у = w.
Finally, let us define a sequence (wt)® of functions by means of the following formula
(3.6) w{ = щ Xsi + ViXS\Si+ zZt.
where i = 1, 2, ... This is the desired sequence of functions. These functions are linearly independent and belong to Cf . Towards the proof of this fact let Уь Уг> •••> Уя denote real numbers and let us consider the linear combination
П
h = £ y, wp., where px < p2 < ... < p„ denote positive integers. Supposing
/i = Owe shall prove that yf = 0 for i = 1, ..., n. The following representation may be verified for the function h:
(3.7) h(co)= <
0, П z(a>) Z Уь
(û$ Sv jT\
(o e T ; i = 1
Z к “* ( « ) + Z yivPi (c°)’ caeSPl \ s Pi+1’
i = 1 i = l+ 1
Z У«МР;И>
i = 1
Z yivPi(°j)’
ы е
П
SPi;i = 1
соф U Sp., coeS.
i = 1 i= 1
First we note that c o e T and h(co) = 0 implies £ yf = 0. Now, using the i
formulas, for w, ub vt given above we obtain for coeSp \ S p :
l n n
h (со) = (и(со)- v (со))(( 1 - у 0) £ У;ар. - у о Z У>-Ръ) + WИ Z У.•
i = 1 i = i + 1 1 = 1
Hence, applying the formula — y0) = (1 — Я,)/?г y0 we get
l n
(3.8) h(co) = ( l - y 0)(n (a ) ) -i> ( < » ) ) ( Z 7iaPi+ Z У/ap<£p()>
i = 1 i = I + 1
where l = 1 and cuGSPj\S Pt+1; we have denoted ep. = — Яр./(1 —
— Яр.); £р.е( —1,0). Note also that y0, ap.g(0, 1), w(w)^i>(a>) on S.
П
Moreover, the same reasoning for m e f ) SPi gives (3.8), with l = n. Collecting all these facts we see that the equality h(co) = 0 for coeS leads to a homogeneous system of equations with respect to yi , . . . , y „ with the determinant:
a p 1 ’ a P 2 fiP2’ a P 3 £ P 3 ’ a Pn £Pn
a PJ ’ a P 2 ’ а Р з £Р з ’ • • • ’ V P n
a p 1 ’ a P 2 ’ a P 3 , . . . ,
a p n h n
a p l ’ a P 2 ’ Otp3 , . . . , a Pn
= ocP l oCp2. . . ccPn( £ p2 - l ) . . . ( £pn - l ) ^ 0 .
Hence yl5 ..., yn must be equal to zero and therefore the proof of the linear independence of the functions wl5 vv2, ... is finished.
Further, the functions defined by (3.6) satisfy (3.3), i.e., (3.9) Ff (wt) = ri, i = l , 2 , . . .
To prove this let us first note that F f (UiXs) = ^ F у(щ) and Ff (ViXs\s)
= Ff{Vi) — Ff (v( Xs) = ( 1— h) Ff (vi). Now we hâve
Ff {Wi) = Ff (щ xs) + F f (v{ Xs\Sj) + F f (Wt) = h F f (щ) + ( 1 - K) F f fo) + F f (z)
= F / (Ai ui + ( l - A i)i?i) + F/ (z) = Ff {w) + Ff (z) = rj, i = 1 , 2 , . . . , where we have applied (3.3) and (3.4).
It remains to prove that property (3.9) is still satisfied for convex combinations of the functions defined by (3.6). For this purpose let px < p2
< ... < p„ denote any fixed positive integers. Also, let y{, where i = 1, 2, ...,
П П
denote real numbers such that £ = 1. Let us denote h = £ yt wp. and let us
i l 1
introduce the following abbreviations:
= A = S p.\S p.+ 1; i = 1, ..., n —1, Dn = SPn.
By virtue of representation (3.7) we obtain the following sequence of equalities.
Ff (n) = Ff ( Y J 7iwp.Xq\t) + Ff(zXt) i = 1
= î Ff ( t ïiWP:XD) + Ff (z) 1=0 i=l
= Î f / [ Œ Vl«„+ Î У,•!>„,)] *D, + f /( z ) /=0 i=l i = 1 +1
= Z ( Z F f ( uPiXd) + Z 7,-F f(vp.&>,)) + (z)
i = 0 i = 1 i = H - 1
= Z У ;(^ /К ,Х 5 р.) + ^ / К Х 5 \ 5 р.)) + ^ /(2 )
1=1 ' 1
n
= Z У . ^ / К гХ«\г) + ^/(^) = »7- i = 1
Thus we have proved that for any real r\ > 0 the infinitely dimensional convex set К = со({и>,-}®) c F J 1^ ) .
In the case where 0 < / ^ + oo on Ar\{0}, / need not be continuous on X. Then, we begin with considering the relation
Г = {(со, x, y ) e Q x X x X : /(со, i( x + y)) = i/(c o , x) + i/( c o , y);
/(со, x),/(co, у) < T-oo; X ^ y}
which has I x J'-measurable graph. To obtain a measurable selector S(co)
= (u1((o), Vt (со))еГ(со) for coedom (r) we must now apply a generalization of Aumann’s selection theorem obtained by M. F. Sainte Beuve (cf. Sur la généralisation d’un théorème de section mesurable de von Neuman-Aumann, C.R.A.S. 276 (1973)). The rest of the proof runs without changes except the part concerning the construction of the function z. Namely, we have now w0
= w for S c: A m with sufficiently small measure if necessary. Thus thesis (a3) of the theorem is proved.
Co r o l l a r y 3.2. Let g be a continuous measure and let f be a Young’s function such that / < + оо ц-а.е. on Q. The following statements are
equivalent.
(a) / is -strictly convex {for fi-a.e. meQ).
(b) Ff is -strictly convex.
(c) Ff is -strictly convex.
Moreover:
(a') I f f does not satisfy f > 0 on X \{ 0 ) {in this case f is not -strictly convex), then for every r ^ 0 there exists a convex set К such that K cz F J 1 (r) and dim(K) ^ N0.
(b') I f 0 < / on X \ { 0 ] and f is not -strictly convex, then for every r > 0 there exists a convex set К such that К c: F J 1^ ) and dim(K) ^ N0.
We recall that ^ - s t r i c t convexity implies ^ - s tr ic t convexity whenever 1 < к < l < + oo.
4. Isotropic Young’s integrands. Given a Young’s integrand /(со, x) that is constant with respect to coeQ, we associate a function tp: X -> R, <p(x)
= /(co , x), coeQ, x e X . This function satisfies the following conditions: (i) tp is l.s.c on X, (ii) <peConv(Y), (iii) q>{0) = 0 and q> ^ 0 on X. Conversely, for a function q> satisfying these conditions one may associate an isotropic Young’s integrand / by means of the formula / (со, x) = (p{x), (со, x ) e Q x X.
Let X be a Banach space not necessarily separable. Let {Q, I , ц) be a measure space that is not purely atomic, i.e., there exists a set S e I such that 0 < H(S) < + oo and S contains no atoms. In the case of isotropic Young’s integrands Theorems 3.1, 3.2 may be now expressed in a weaker form.
Th e o r e m 4.1. Suppose q>: X —► R + is a function satisfying the above- mentioned conditions (i), (ii), (iii). Let for u e M x {Q) F^(u) = J (p(u{(o))dp.
Q
(a) Then, Fv is J ^ 1-strictly convex if and only if (p is .jè1-strictly convex. I f (p is not ZF1 -strictly convex then Fv is not -strictly convex. Moreover:
(bl) I f (p{x) — 0 fo r some x Ф 0, then there exists an infinitely dimensional convex set К contained in F ~ 1 (r), where r — 0.
(b2) I f q> > 0 on X \{0}, but q) is not J^ 1-strictly convex, then for each sufficiently small r > 0 there exists an infinite-dimensional convex set К contained in F ÿ 1^).
R e m a r k . If, moreover, r - > + o o implies q>(nc)-++oo for 0 # x e l , then in case (Ь2) r > 0 may be taken arbitrarily. Also, in case (bl) one may put any r > 0 whenever the following function r -><p(rx) is continuous as a numerical function.
A part of the above theorem has been proved in [6], for X = R 1, with a slight different construction of the set K. Here, the above theorem may be proved by usage of the method applied in the proofs of Theorems 3.1, 3.2 with suitable simplifications. Indeed, we now need not work with selectors of multifunctions.
Finally, let us mention that a problem arises to characterize integrands / admitting extremely large fases on epi(Ff ). We point out that if / = (p (isotropic case of / ) is homogeneous function with respect to zero, then (p cannot be ^ -strictly convex, even if it is finite on X. Therefore, the functional Fv is not J 5"1-strictly convex. Moreover, let 0 < < p < + o o on X \{0}. Then q>(x) = ||x||x, and for every r > 0 F l \ (r) contains an infinitely dimensional convex set. On the other hand Тц.ц defines a norm on the space L X(Q, A) of Bochner integrable functions contained in M X(Q) (after identifi
cation of functions being equal each other /л-а.е. on Q). We have L x (ü, R 1) c* L x (Q, X) isometrically. It is known, cf. [8], that L x (Q, R 1) is a flat space. Thus L x (Q, X) is also flat. This implies that the unit sphere S in L X(Q, X) (we have S = (1)) contains a flat area that is the largest possible in the sense of diameter. This area is not weakly compact. Thus, we have obtained in the case under consideration: for every r > 0 there exists a convex set К such that K c y (r), diam {K) = 2-r = diam Fj[.{t (r) and dim (К ) ^ N0.
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