DENTABLE FUNCTIONS AND RADIALLY UNIFORM QUASI-CONVEXITY

. In this paper we give a further result which states sufficient conditions for the theory of convergence of minimizing sequences to be applicable, develop the theory further, and give an application

C. G. LOONEY minimizing sequence of f on C will converge in norm to a minimum (when it exists).
In this paper we give a further result which states sufficient conditions for the theory of [4] to be applicable, develop the theory further, and give an application.

SOME DEFINITIONS AND THEOREMS
The closed convex hull of a set S in X is denoted by cl-conv(S).We say S is dentable at x e S whenever: given any e > 0, x cl-conv (SB (x)), where B (x) is the open e-ball centered on x.In this case x is called a denting point of S [5] (strongly extremal point of S, [2; p. 97]).See [9] for the origin of the term "dentable".
A function f on a convex set C is said to be dentable [4] at x 0 e C iff (x 0, f(x0)) is a denting point of epi(f) {(x,) X R --> f(x)}.It was shown in [4] that if f is a 1.s.c.(lower semi-continuous) quasi-convex functional on a weakly compact convex set C and has a unique minimum x 0 C, then every minimizing sequence of f converges in norm to x 0 iff f is dentable at x 0. We say that f is quasi-convex on C iff the level sets L {x e C: f(x) =< are convex.This is equivalent to the following: for any x, y e C, f(%x + (i -%)y) <= max {f(x), f(y)}, 0 =< % < i. Convex functionals are quasi- convex, but not conversely.
A normed space X, its closed unit ball, and its norm are all said to be uniformly convex iff given e > 0 and x, y with lxll =< i, lYll -< I and II x Yll > e, there exists () > 0 such that l1/2x + Yll -< I ().Every such space is strictly convex, i.e., l%x + (I %)Yll < %11xl + (I %)IIYlI, 0 < % < I.It is shown [3] that the p and L spaces are uniformly convex P for < p < .When the modulus of convexity depends on the point x also, i.e., (x,) > 0, then we say that II is locally uniformly convex [8] Locally uniformly convex spaces are not generally uniformly convex, but the converse is true.Also, locally uniformly convex spaces are strictly convex, but not conversely.
THEROEM i.If X is a locally uniformly convex space, then every boundary point of B I(0) is a denting point of B I(0). PROOF.Let x have norm i.Given e > 0, let Qe BI(0)B (x).For any Y e Qg, flY xll > so that there is some (x,e) > 0 such that I11/2 x + 1/2 Yll i 6.The set BI_(0) can be strictly separated from x by a closed hyper- plane (see, e.g., [3; p.193]) H which partitions X into two halfspaces H I and H 2 with H I being closed and containing Qe. H 2 contains x as an interior point.Then cl-conv (Q)HI closed convex set, so that x cl-conv (Qs).
LEMMA I. Let C be a compact convex subset and f: C R be l.s.c, and have a unique minimum x 0 e C. Then for any e > 0, f is bounded away from B f(x0) on

CB
). g (X O PROOF.The set Q CB (Xo) is compact and thus f attains its infSmum y on Q.By hypothesis, y > B. Thus f(x) -> y > B on Q. CONSTRUCTION.Let f assume its infimum at a unique point x 0 e C closed bounded convex subset of a normed space X that is locally uniformly convex.The set H {(x,B): x e X} meets epi(f) at the point (x0,B).Let L be the vertical line {(x0,): e R} in X x R. Fix r and take the point Pr (x 0 + r) e L at a distance r above (x 0 ).Let B (Pr) be the closed ball of radius r centered r on Pr" Then Br(Pr meets H at the unique point (x 0,8).
Let f be a l.s.c, functional on a convex bounded set C and let f have a unique minimum x 0 e C. If X is a locally uniformly convex normed space, then a sufficient condition for f to be dentable at x 0 is that either, i) there exist some r > 0 such that Er be contained in Br(Pr )' or ii) C be compact.
ii) For any e > 0, f is bounded away from f(x 0) on CBe(x0).Thus f(x 0) is seprated by a closed hyperplane from epi(f)(Be(x0, f(x0) The conditions of Theorem 2 part i) are srong, and as the next example shows, may not be satisfied even in a finite dimensional space. 2 -I/x EXAMPLE i.Let f(x) e for x # 0 and f(0) 0. Then f is a con- tinuous quasi-convex function R, but we take C to be the compact set [-I, I].
The point x 0 is the unique minimum.Let Br(Pr be centered on Pr (0,r) R2.The bottom branch of the sphere of B (pr) is a convex function g(x) in r r x We show now that llm f(x)/g(x) 0, so that f(x) < g(x) on some nbhd of 0. It can be shown that f' (0) 0 f"(0), by putting t I/x and using the limit definition for derivatives at x 0. Then lim f(x)/g(x) lim -1/2 f" (x) / g" (x) 0/(r2) (by use of L 'Hospital's Rule twice).Thus for any given r, there is some e > 0 such that epi(f) is not contained in B (pr) for -< x < r Although f does not satisfy the hypothesis of Theorem 2, it is dentable by part ii of Theorem 2.
We note that Theorem 2i can be phrased in terms of the Gteaux derivatives of f and g, where g(x) inf {e:(x,e) r(Pr)}, i.e., we need ]f'(x;y) => ]g'(x;y) at x for all y C.
3. NEW TYPE OF QUASI-CONVEXITY AN IMPORTANT APPLICATION.
We say that a subset S of a vector space is radially convex at s O S iff any line L through s o meets S in a convex set LS. Any convex set C is radially convex at any point of C.
A functional f defined on S is said to be, respectively, radially convex, radially, strictly co.n.vex, radially uniformly convex, or radially locally uni- forml convex at s o S, whenever S is radially convex at s O and f is, reap., convex, strictly convex, uniformly convex, or locally uniformly convex on all segments LS through s 0. We replace "convex" by "quasi-convex" to get the resulting four new definitions for f at s o in the radially convex set S.
We now consider the space C[e,8] of all continuous functions on the inter- val Is,8] and the subset of rational functions.It is known(see [I] or [7]) THEOREM 3. The functional g + lgll iS radially uniformly quasl-convex at 0 on any convex nbhd U of 0 in any normed space.
We note that a norm may not be radially strictly convex, e.g., II'II and II-II 1 are linear on line segments from 0 to any x(x must be in the positive is uniformly convex and orthant in the case of II IIi )-For 1 < p < l'llp therefore is radially strictly convex on any radially convex set w.r.t, to a point.
THEOREM 4. The function a + lf (a 0 + + a xm) ll is radially uni- m p formly quasi-convex at the minimum a (a 0,...,am) on any closed bounded nbhd of a for f fixed in C [e,8], < p < PROOF.Without loss of generality, we assume that f 0. The linear function a / (a 0 +...+ am xm) followed by the convex map If" II is trivially Rm+l convex with minimum on at 0. Since the space of polynomials P [e,8] of m degree m or less is linearly isomorphic to R m+l a / lla 0 +...+ a xmll is m p actually a norm on and thus Theorem 2 holds (it is known that a unique minimum a does indeed exist for 1 < p).
We now borrow a proposition from [7].See [I] also.
COROLLARY 2. Starting from any point (a0, b0) # (a*, b*) there is a direction d o in which T, decreases.Further, given a step size of eO there 0 dO is a 0 > 0 such that T will decrease by at least along until the directional minimum is reached., COROLLARY 2. T is dentable at its unique minimum (a, b), and any mini- , mizing sequence (an, bn) converges to (a, b the approximation functional T (a,b) II f(') r (a b;-) II is quasi- /(b 0 + + b xn).The norm II'II is not strictly convex since its m n graph on the unit ball contains horizontal llne segments.