Extreme and Exposed Points in Orlicz Spaces

Abstract

Extreme points of the unit sphere in any Orlicz space over a measure space that contains no atoms of infinite measure are characterized. In the case of a finite-valued Orlicz function and a nonatomic measure space, exposed points of the unit sphere in these spaces are characterized too. Some corollaries and examples are also given.

Orlicz spaces without extreme points

“…Although methods developed in papers [10,11,13] are useful in our considerations, in many cases to solve the problems from this paper, developing of some new techniques was necessary.…”

A full description of extreme points in 𝐶(Ω,𝐿^{𝜙}(𝜇))

“…every but one) t £ T (cf. [5,16]). Thus the only extreme point of B(L9(p)) with x > 0, I (x) < 1 can be the function e defined by e(t) = c(cp) for t £ T. Let us fix an arbitrary element co0 of the set W. We have e £ extB(L<p(p)) iff \\e\\ > 1 (indeed, if ||e|| < 1 then c(cp) < co and <p(c(<p)) < oo; so (c(tp), <p(c(cp))) are points of strict convexity of Graph cp , and by virtue of [5,16], e £ tx\B(L9(p))).…”

Orlicz Spaces Without Strongly Extreme Points and Without H-Points

“…t G T. Next by the finiteness of /i we have Iq>(\x) < 00 for any A > 0. This is a contradiction, because in the case when I®(x) = 0 the quality ||JC||<D = 1 implies that I®(\x) -00 for any À > 1 (see [2]).…”

“…PROOF. It is known that every strongly extreme point is extreme and that under the assumptions concerning O, if x G 5(L°) is extreme then it must be I®(x) -1 (see [2]). Therefore, it suffices to consider only these points of 5(L°) for which I®(x) = 1.…”