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We give a general result on the behavior of spreading models in Banach spaces which coarse Lipschitz-embed into asymptotically uniformly convex spaces. We use this result to study the uniqueness of the uniform structure in ℓ p \ell _p -sums of finite-dimensional spaces for 1 > p > ∞ 1>p>\infty ; in particular we give some new examples of spaces with unique uniform structure.

Two examples of asymptotic ℓ 1 \ell _{1} Banach spaces are given. The first, X u X_{u} , has an unconditional basis and is arbitrarily distortable. The second, X X , does not contain any unconditional basic sequence. Both are spaces of the type of Tsirelson’s.

We present a second order differentiability theory for convex functions on Banach spaces.

Generalized second-order derivatives introduced by Rockafellar in finite-dimensional spaces are extended to convex functions in reflexive Banach spaces. Parallel results are shown in the infinite-dimensional case. A result that plays an important role in applications is that the generalized second-order differentiability is preserved under the integral sign.

The present paper is an approach to the local theory of Banach spaces via the ultrapower construction. It includes a detailed study of ultrapowers and their dual spaces as well as a definition of a new notion, the notion of a u-extension of a Banach space. All these tools are used to give a unified definition of many classes of Banach spaces characterized by local properties (such as the L p {\mathcal {L}_p} -spaces). Many examples are given; also, as an application, it is proved that any L p {\mathcal {L}_p} -space, 1 > p > ∞ 1 > p > \infty , has an ultrapower which is isomorphic to an L p {L_p} -space.

For a convex Orlicz functionφ:R+→R+∪{∞}\varphi :{{\bf {R}}_ + } \to {{\bf {R}}_ + } \cup \{ \infty \}and the associated Orlicz sequence spacelφ{l_\varphi }, we consider the following five properties: (1)lφ{l_\varphi }has a subspace isometric tol1{l_1}. (2)lφ{l_\varphi }is Schur. (3)lφ{l_\varphi }has normal structure. (4) Every weakly compact subset oflφ{l_\varphi }has normal structure. (5) Every bounded sequence inlφ{l_\varphi }has a subsequence(xn)({x_n})which is pointwise and almost convergent tox∈lφx \in {l_\varphi }, i.e.,limsupn→∞∥xn−x∥φ>liminfn→∞∥xn−y∥φ\lim \,{\sup _{n \to \infty }}\parallel {x_n} - x{\parallel _{\varphi }} > \lim \inf _{n \to \infty }\parallel {x_n} - y{\parallel _\varphi }for ally≠xy \ne x. Our results are: (1)⇔φ\Leftrightarrow \;\varphiis either linear at0(φ(s)/s=c>0,0>s⩽t)0\;(\varphi (s)/s = c > 0,0 > s \leqslant t)or does not satisfy theΔ2{\Delta _2}-condition at00. (2)⇔lφ\Leftrightarrow \;{l_\varphi }is isomorphic tol1⇔φ′(0)=limt→0φ(t)/t>0{l_1}\; \Leftrightarrow \;\varphi ’(0) = {\lim _{t \to 0}}\,\varphi \,(t)/t > 0. (3)⇔φ\Leftrightarrow \varphisatisfies theΔ2{\Delta _2}-condition at0,φ0, \varphiis not linear at00andC(φ)=sup{φ(t)>1}>12C(\varphi ) = \sup \,\{ \varphi \,(t) > 1\} > \frac {1}{2}. (4)⇔φ\Leftrightarrow \,\varphisatisfies theΔ2{\Delta _2}-condition at00andC(φ)>12orφ′(0)>0C\,(\varphi ) > \frac {1}{2}\;{\rm {or}}\;\varphi ’(0) > 0. (5)⇔φ\Leftrightarrow \;\varphisatisfies theΔ2{\Delta _2}-condition at00andC(φ)=1C(\varphi ) = 1. The last equivalence contains a result of Lami-Dozo [10].

A Banach space is flat if the girth of its unit ball is 4 and if the girth is achieved by some curve. (Equivalently, its unit ball can be circumnavigated along a centrally symmetric path whose length is 4.) Some basic geometric properties of flat Banach spaces are given. In particular, the term flat is justified.

An important theorem of Kakutani and Mackey characterizes an infinite dimensional real (complex) Hilbert space as an infinite dimensional real (complex) Banach space whose lattice of closed subspaces admits an orthocomplementation. This result, also valid for quaternionic spaces, has proved useful as a justification for the unique role of Hilbert space in quantum theory. With a like application in mind, we present in the present paper a number of characterizations of real and complex Hilbert space in the class of locally convex spaces. One of these is an extension of the Kakutani-Mackey result from the infinite dimensional Banach spaces to the class of all infinite dimensional complete Mackey spaces. The implications for the foundations of quantum theory are discussed.

Two properties of B-convexity are shown to hold for P-convexity: (1) Under certain conditions, the direct sum of two P-convex spaces is P-convex. (2) A Banach space is P-convex if each subspace having a Schauder decomposition into finite dimensional subspaces is P-convex. 0. Introduction. In the previous paper [1] the question of whether all B-convex spaces are reflexive was discussed. The concept of a P-convex space was introduced by C. Kottman [4] as follows: Definition. For a positive integer n, let P(n, X) be the supremum of all numbers r such that there is a set of n disjoint closed balls of radius r inside U(X)= fx: llxll < 11. X is said to be P-convex if P(n, X) < Y2 for some n. Kottman showed that all P-convex spaces are both B-convex and reflexive. Therefore the question "Is there a B-convex space that is not P-convex?" is of interest. Many properties of B-convex spaces are not known for P-convex spaces. In this paper we consider two of these properties and prove partial analogs of them for P-convex spaces: The first property is that direct sums of B-convex spaces are B-convex [2]. The proof of this fact for B-convex spaces rests on the invariance of B-convexity under isomorphism, but it is not known whether P-convexity possesses this invariance. Two partial analogs of the direct sum property are obtained, Theorems 1.3 and 1.5, using Ramsey's theorem of combinatorics. The second property is that a space is B-convex if each subspace having a basis is B-convex [1]. A partial analog of this is proved, Theorem 2.1, using one of the direct sum results. We will use the following characterization of P-convexity from Remark 1.4 of [4]: Let a set of n elements be called 5 separated of order n provided the distance between any two elements of the set is at least &. Then a space X is P-convex if and only if for some positive integer n and some positive number e < 2 there is no 2 e separated set of order n in U(X). Received by the editors July 7, 1971. AMS (MOS) subject classifications (1970). Primary 46B10; Secondary 46B05, 46B15, 05A05.

Given two Banach spaces F | | F|| and X | | | | X||\,|| , write F fr X iff F{\text { fr }}X{\text { iff}} for each finite-dimensional subspace F ′ F’ of F F and each number ε > 0 \varepsilon > 0 , there is an isomorphism V V of F ′ F’ into X X such that | | x | − | | V x | | | ≤ ε ||x| - ||Vx||| \leq \varepsilon for each x x in the unit ball of F ′ F’ . Given a property P {\mathbf {P}} of Banach spaces, X X is called super- P iff F fr X {\mathbf {P}}{\text { iff }}F{\text { fr }}X implies F F is P {\mathbf {P}} . Ergodicity and stability were defined in our articles On B B -convex Banach spaces, Math. Systems Theory 7 (1974), 294-299, and C. R. Acad. Sci. Paris Ser. A 275 (1972), 993, where it is shown that super-ergodicity and super-stability are equivalent to super-reflexivity introduced by R. C. James [Canad. J. Math. 24 (1972), 896-904]. Q Q -ergodicity is defined, and it is proved that super- Q Q -ergodicity is another property equivalent with super-reflexivity. A new proof is given of the theorem that J J -spaces are reflexive [Schaffer-Sundaresan, Math. Ann. 184 (1970), 163-168]. It is shown that if a Banach space X X is B B -convex, then each bounded sequence in X X contains a subsequence ( y n ) ({y_n}) such that the Cesàro averages of ( − 1 ) i y i {( - 1)^i}{y_i} converge to zero.

Let E E be a Banach space. One often wants to measure how far E E is from being a Hilbert space. In this paper we define the skewness s ( E ) s(E) of a Banach space E E , 0 ⩽ s ( E ) ⩽ 2 0 \leqslant s(E) \leqslant 2 , which describes the asymmetry of the norm. We show that s ( E ) = s ( E ∗ ) s(E) = s({E^{\ast }}) for all Banach spaces E E . Further, s ( E ) = 0 s(E) = 0 if and only if E E is a (real) Hilbert space and s ( E ) = 2 s(E) = 2 if and only if E E is quadrate, so s ( E ) > 2 s(E) > 2 implies E E is reflexive. We discuss the computation of s ( L p ) s({L^p}) and describe its asymptotic behavior near p = 1 , 2 p = 1,2 and ∞ \infty . Finally, we discuss a higher-dimensional generalization of skewness which gives a characterization of smooth Banach spaces.

The supremum of the ratios of the self-radiusrA(A){r_A}(A)of a convex bounded set in a normed linear spaceXXto its absolute radiusrX(A){r_X}(A)is related to the supremum of the relative projection constants of the maximal subspaces ofXX. Necessary conditions and sufficient conditions for these suprema to be smaller than 2 are given. These conditions are selfadjoint superproperties similar toBB-convexity, superreflexivity andPP-convexity.

It is proved that: (i) every complemented subspace in an infinite product of L 1 {L_1} -predual Banach spaces ∏ i ∈ I X i \prod \nolimits _{i \in I} {{X_i}} is isomorphic to Z × K m Z \times {{\mathbf {K}}^\mathfrak {m}} , where dim ⁡ K = 1 , m ⩽ card ⁡ I \dim {\mathbf {K}} = 1,\;\mathfrak {m} \leqslant \operatorname {card} I and Z Z is isomorphic to a complemented subspace of ∏ i ∈ J X i , J ⊆ I , Z \prod \nolimits _{i \in J} {{X_i},\;J \subseteq I,\;Z} contains an isomorphic cop[ill] of c 0 card ⁡ J c_0^{\operatorname {card} J} ; (ii) every injective lcs (in particular, Fréchet) is of the form Z × K m , dim ⁡ K = 1 Z \times {{\mathbf {K}}^\mathfrak {m}},\;\dim {\mathbf {K}} = 1 , where Z Z has a fundamental family of seminorms of the cardinality τ \tau and Z Z contains an isomorphic copy of l ∞ r l_\infty ^\mathfrak {r} (this is a generalization of Lindenstrauss’ theorem on injective Banach spaces); (iii) whenever X ≃ l p , 1 ⩽ p ⩽ ∞ X \simeq {l_p},\;1 \leqslant p \leqslant \infty , or X ≃ c 0 X \simeq {c_0} , then every complemented subspace in a power X m {X^\mathfrak {m}} ( m \mathfrak {m} is an arbitrary cardinal number) is isomorphic to X r × K s , r + s ⩽ m {X^\mathfrak {r}} \times {{\mathbf {K}}^\mathfrak {s}},\;\mathfrak {r} + s \leqslant \mathfrak {m} (a generalization of the results due to Lindenstrauss and Pełczyński for m = 1 \mathfrak {m} = 1 ).

Necessary and sufficient conditions for a positive operator to fac- tor through a Banach lattice satisfying upper and lower estimates are pre- sented. These conditions are then combined to give a necessary condition for a positive operator to factor through a super-reflexive Banach lattice. An ex- ample is given to show that, in spite of the name given by Beauzamy, uniformly convexifying operators need not factor through any uniformly convex lattice 0. Introduction. In contrast to the elegant equivalence between super-reflexive Banach spaces and spaces that can be renormed with an equivalent uniformly convex norm, worked out by James and Enflo, the results on operators is not as satisfying or complete. Beauzamy (1) restated some of James' definitions for super-reflexity in terms of operators. The operators he defined to be uniformly convexifying, however, may not even factor through any uniformly convex space. Possibly a better name would be super-reflexive maps since any operator that is finitely representable in a uniformly convexifying operator cannot have the Finite Tree Property. Davis, Figiel, Johnson, and Pelczyhski (4) have shown that weak compact maps are "reflexive" in the sense that they factor through reflexive spaces. However, there are no known conditions on T: X —> Y which are equivalent to T being factorable through a uniformly convex space. In this paper, necessary and sufficient conditions are obtained if X and Y are Banach lattices, T positive, and the factorization uses positive operators. 1. Preliminaries. In this section definitions and results that will be used throughout this paper will be given. Notation will generally be consistent with Lindenstrauss and Tzafriri (9). A partially ordered normed space X over the reals is called a normed lattice provided

We study intersection properties of balls in Banach spaces using a new technique. With this technique we give new and simple proofs of some results of Lindenstrauss and others, characterizing Banach spaces with L 1 ( μ ) {L_1}(\mu ) dual spaces by intersection properties of balls, and we solve some open problems in the isometric theory of Banach spaces. We also give new proofs of some results of Alfsen and Effros characterizing M-ideals by intersection properties of balls, and we improve some of their results. In the last section we apply these results on function algebras, G-spaces and order unit spaces and we give new and simple proofs for some representation theorems for those Banach spaces with L 1 ( μ ) {L_1}(\mu ) dual spaces whose unit ball contains extreme points.

John R. Rice [Approximation of functions. Vol. II, Addison-Wesley, New York, 1969] investigated best approximation from a nonlinear manifold in a finite dimensional, smooth, and rotund space. The authors define the curvature of a manifold by comparing the manifold with the unit ball of the space and suitably define the “folding” of a manifold. Rice’s Theorem 11 extends as follows: Theorem. Let X be a uniformly smooth Banach space, and F : R n → X F:{R^n} \to X be a homeomorphism onto M = F ( R n ) M = F({R^n}) . Suppose ∇ F ( a ) \nabla F(a) exists for each a in X, ∇ F \nabla F is continuous as a function of a, and ∇ F ( a ) ⋅ R n \nabla F(a) \cdot {R^n} has dimension n. Then, if M has bounded curvature, there exists a neighborhood of M each point of which has a unique best approximation from M. A variation theorem was found and used which locates a critical point of a differentiable functional defined on a uniformly rotund space Y. [See M. S. Berger and M. S. Berger, Perspectives in nonlinearity, Benjamin, New York, 1968, p. 58ff. for a similar result when Y = R n Y = {R^n} .] The paper is concluded with a few remarks on Chebyshev sets.

A proof of James that uniformly nonsquare spaces are reflexive is extended in part to B-convex spaces. A condition sufficient for non-B-convexity and related conditions equivalent to non-B-convexity are given. The following theorem is proved: A Banach space is B-convex if each subspace with basis is B-convex. 0. Introduction. The notion of a B-convex Banach space was introduced by A. Beck [1], [21 as a characterization of those Banach spaces X having the property that a certain strong law of large numbers holds for X valued random variables. Definition. Let k be a positive integer and e a positive number. X is said to be k, t-convex if for any 1x1, ...* Xkl, Kxi 1 < 19 i = 1, * * * , k, there is some choice of signs 61' * * k so that xk_ 1| < k(l c). X is said to be B-convex if it is k, t-convex for some k and c. Further study of B-convex spaces has been done by R. C. James [61, [71, D. P. Giesy [51 and C. A. Kottman [8]. Giesy showed that B-convex spaces have many of the properties of reflexive spaces. James conjectured that all B-convex spaces are reflexive, and proved the conjecture true for 2, E-convex spaces. Both James and Giesy proved the conjecture true for B-convex spaces having an unconditional basis. Kottman extended James' 2, E-convex proof to a larger subclass, P-convex spaces. Examples are known of spaces which are reflexive but not B-convex. ?1 of this paper adopts a part of James' 2, E-convex theorem to all non-B-convex spaces, presents a condition sufficient for non-B-convexity, and gives related characterizations of non-B-convex spaces, though the conjecture of James remains open. ?2 proves a theorem on B-convexity and subspaces with basis analogous to a theorem of Pekczyn'ski on reflexivity and subspaces with basis. For a Banach space X, U(X) will denote the closed unit ball lx: |lxii < 11 of X. I. Non-B-convexity. In James' proof [6] that 2, E-convex spaces are reflexive, he defines for a Banach space X a sequence of numbers Kn, and shows that if X Received by the editors July 7, 1971. AMS (MOS) subject classifications (1970). Primary 46B10; Secondary 46B05, 46B15.

The direct product Ei®nE2 of two Banach spaces Eu E2 has been defined before [5](2) as the closure of the normed linear set $In(Ei, E2) (that is, linear set 3i(£i, £2) of expressions 22Â-ifi®4>i, hi which N is a norm) [5, p. 200, Definition 1.3] and [6, p. 499, b]. Let N denote a crossnorm whose associate N' is also a crossnorm [5, p. 208]. Then, the cross-space Ei®nEi determines uniquely a "conjugate space" (Ei®NE2)' and an "associate space" Ei ®n'E2 . It is shown [5, p. 205] that Ei ®N'E2 is always included in (Ei®nE2)'. While there are many known examples of cross-spaces for which the associate space coincides with the conjugate space—for example, the cross-space generated by the self-associate crossnorm constructed for Hubert spaces by F. J. Murray and John von Neumann [3, p. 128] and [5, pp. 212-214]—it is not without interest to construct a cross-space for which the associate space forms a proper subset of the conjugate space (§§1-2). For reflexive Banach spaces Ei, E2 (that is, such that 25/' =Ei), and a reflexive crossnorm N [6, p. 500], the reflexivity of Ei®^E2 implies (Ei®nE2)'—Ei ®N'Ei [6, p. 505]. Thus, the finding of the exact conditions imposed upon reflexive Banach spaces and a reflexive crossnorm for which the resulting cross-space is reflexive is closely connected with the above-mentioned problem. In §1, we show that for a "natural crossnorm" N, L'®nL' is a proper subset of (L®NL)'. In §2 we prove that for a "natural crossnorm" N, V®ud' is a proper subset of (/ ® nI) '• In §3 we show that for any p > 1, lp ® nIi is not reflexive, provided l/p + l/q = 1 and N denotes the least crossnorm whose associate is also a crossnorm [5, p. 208]. The last one is reflexive [6, p. 501 ]. 1. Let Z,(i) and L$) denote the Banach spaces of all functions integrable in the sense of Lebesgue on the interval Ogsgl, and on the square 0^s, t ^ 1 respectively. Similarly, let M(u and M denote the Banach spaces of all functions Lebesgue measurable and essentially bounded on the interval OiSs^l and the square 0^s, f gl respectively [l, pp. 10, 12]. We recall that

Introduction. In a previous paper [7](1), for two Banach spaces E1, E2, the Banach spaces E1?E2, E; X E2, E" G E2 ' [7, p. 205 ] are constructed. If the norm N [7, Definition 3.1] is defined on E1?E2, then the associate norm N' [7, Definition 3.2 and Lemma 3.11 is defined on E; X E2. Similarly N" denotes the norm on EP X E2'. Among the unsolved problems (mentioned in [7, ?6]), are listed the following two: A. What are the exact conditions imposed upon a crossnorm [7, Definition 3.3] under which (E10 E2)'=E; XE2' holds? B. rs the associate with every crossnorm also a crossnorm, or do there exist crossnorms whose associates are not crossnorms? In the present paper we present a "partial" answer to problem A (which we denote by A*), and a "partial" answer to problem B (which we denote by B*). A*. A uniformly convex crossnorm N sets up the relation (E1l0E2)' E' 0EX if, and only if, N"= N. B*. For reflexive Banach spaces (that is, such that EP = E1, E' =E2) the associate with every crossnorm is also a crossnorm. In this paper we also show that the values of a crossnorm for all expressions of rank not greater than 2 do not necessarily determine the crossnorm. The following should be mentioned in immediate connection with problem A: It is evident that for norms for which (E1l E2)'= E( XE' holds, N" = N. Since in general (for any norm N) all we can state is (E1l?E2)'DE; ?El [7, p. 205], we have no basis for assuming that N" represents the norm in (E1l E2)", or N" =N for expressions in (E1, E2)C2[(E', E2') [7, Definition 1.3]. Therefore, N"

D(V) of V. In the rest of the paper let E[r] and E' stand for an l.c. space and its dual. In Dixmier's characterization of conjugate Banach spaces, an essential role was played by subspaces of characteristic one in the dual. In the generalization to l.c. spaces mentioned above, the analogous role was taken by what we called duxial subspaces, defined as follows. A linear subspace V of E' is said to be duxial if every convex 7-r(E)-compact subset of E' is contained in the 7-r(E)-closure of a bounded subset of V[-rbi(E, E')], that is, V with the topology induced by E'[rb(E)]. It is easy to see that a duxial subspace in the dual of a Banach space is nothing but a subspace of characteristic greater than zero.

Recently, in these Transactions, Nachbin [N] and, independently, Goodner [G] have shown that if B has the extension property and if its unit sphere has an extreme point, then B is equivalent to a function space of this sort; both authors have also proved that such a function space has the extension property. The above theorem simply omits the extreme point hypothesis, and so establishes the equivalence. My original proof, of which the proof given here is a distillate, depends on an idea of Jerison [j]. Briefly, letting X be the weak* closure of the set of extreme points of the unit sphere of the adjoint B*, B can be shown equivalent to the space of all weak* continuous real functions / on X such that/(x) = —f( — x), and then properties of X are deduced which imply the theorem. The same idea occurs implicitly in the proof below. Note. Goodner asks [G, p. 107] if every Banach space having the extension property is equivalent to the conjugate of an abstract (L)-space. It is known (this is not my contribution) that the Birkhoff-Ulam example ([B, p. 186] or [HT, p. 490]) answers this question in the negative, the pertinent Banach space being the bounded Borel functions on [0, 1 ] modulo those functions vanishing except on a set of the first category, with ||/|| = inf {K: \f(x) | g K save on a set of first category}. 1. Preliminary definitions and remarks. A point x is an extreme point of a convex subset K of a real linear space if x is not an interior point of any line segment contained in K (i.e., if x=ty + (l — t)z, 0

If B B is l p ( ω ) {l_p}(\omega ) or c 0 ( ω ) {c_0}(\omega ) we show B B has the following extension property. Any homeomorphism from a compact subset M M of B B into B B may be extended to a homeomorphism of B B onto B B which is a C ∞ {C^\infty } diffeomorphism on B ∖ M B\backslash M to its image in B B . This is done by writing B B as a direct sum of closed subspaces B 1 {B_1} and B 2 {B_2} both isomorphically isometric to B B so that the natural projection of K K into B 1 {B_1} along B 2 {B_2} is one-to-one (see H. H. Corson, contribution in Symposium on infinite dimensional topology, Ann. of Math. Studies (to appear)). With K , B , B 1 K,B,{B_1} and B 2 {B_2} as above a homeomorphism of B B onto itself is constructed which leaves the B 1 {B_1} -coordinates of points in B B unchanged, carries K K into B 1 {B_1} and is a C ∞ {C^\infty } diffeomorphic map on B ∖ K B\backslash K . From these results the extension theorem may be proved by standard methods.

which we shall call local uniform convexity. Geometrically this differs from uniform convexity in that it is required that one end point of the variable chord remain fixed. In section I we prove a general theorem on the product of locally uniformly convex Banach spaces and with the aid of this theorem we establish that the two notions are actually different. Section II is devoted to the investigation of the relationship between local uniform convexity and strong differentiability of the norm. In section III we investigate conditions for isomorphism of a Banach space with a locally uniformly convex space.

A measure of the “massiveness” of the unit ball of a Banach space is introduced in terms of an efficiency of the tightest packing of balls of equal size in the unit ball. This measure is computed for the l p {l_p} -spaces, and spaces with distinct measures are shown to be not nearly isometric. A new convexity condition, which is compared to B B -convexity, uniform smoothness, and uniform convexity, is introduced in terms of this measure, and is shown to be a criterion of reflexivity. The property dual to this convexity condition is also exposed and examined.

Introduction. This paper contains the first unified treatment of the dual theory of differentiability of the norm functional in a real normed linear space. With this, the work of Smulian [2; 3] is extended and it is shown how uniform convexity is to be modified so as to obtain geometric properties dual to the various types of differentiability of the norm thus answering a question implicit in the work of Lovaglia [1] and Anderson [1]. The resulting dual theory of differentiability is then used to obtain a connection between the differentiability of certain infinite dimensional manifolds imbedded in an infinite dimensional real normed linear space and the continuity properties of the Gaussian spherical image map generalized to such infinite dimensional spaces. In this way a problem proposed by Klee [2, p. 35] is solved. The principal tools employed are (a) Mazur's characterization [1] of a supporting hyperplane of the unit ball as an inverse image of the derivative of the norm functional; (b) a modification of an integral calculus technique given in Krasnosel'skir and RutickiT [1, p. 187]; and (c) James' criterion for the reflexivity of a Banach space, viz., that every continuous linear functional attains its supremum on the unit ball. James [1] has given a proof of this criterion for separable Banach spaces and in James [2] has removed the condition that the Banach space be separable. The paper is divided into five sections. ?1 contains the localization and directionalization of uniform convexity of Clarkson [1] and of full k-convexity of Fan and Glicksberg [1]. ?2, while containing interesting facts in its own right, is motivational in nature for ?3. ?3 contains in the dual theory of differentiability of the norm one of the two main conclusions of the paper. The other main conclusion contained in ?4 is the analysis of the differentiability of norm functionals using the generalized Gaussian spherical image map. The last section, ?5, examines and compares the present results in the context of the geometry of Banach spaces as created and perfected by other workers in the field.

Introduction. The purpose of this paper is to study a convexity property on normed linear spaces (NLS's) which we call B-convexity. Interest in B-convexity is generated by a theorem of Anatole Beck ([1] or [2]) which states that a Banach space X is B-convex if and only if a certain strong law of large numbers is valid for X-valued random variables. Let X be a NLS and (S, E, m) a measure space. The Borel a-field 9 of X is the a-field of sets generated by the subsets of X open in the strong (norm) topology. A Borel set is an element of M. A function X from S into X is called strongly measurable if for each B E M, X 1(B) = {s E S: X(s) E B} E S. X is called essentially separably valued if there is N e E, m(N) =0, such that X(SN) is separable. If fs 11X(s) 1 dm(s) X which is strongly measurable and essentially separably valued. The expectation of X, E(X)= fQ XddY if this integral exists. For a random variable X with expectation, we define the variance of X by

0. Introduction. Let X be a compact convex set and let F be a closed face of X. In this paper we develop a technique which yields sufficient conditions for F to be a peak-face of X (a subset of X where a continuous affine function on X attains its maximum). The theory is based on a duality between certain types of ordered Banach spaces. This duality is an extension of the results of [6] (see also [17]) and relates the directness of an ordered Banach space E to the degree to which the triangle inequality can be reversed on the positive elements of E*. A precise formulation of this is given in ?1. In ?2 we define a compact convex set X to be conical at an extreme point x if there is a bounded nonnegative affine function f on X such that f(x) =0 and X= conv ({x} u {y E X: f(y) > 1}). If X is conical at the G, extreme point x then the results of ?1 are applied to show that x is a peak-point of X. Every compact convex set Xhas a natural identification with the positive elements of norm one in A(X)*, where A(X) is the space of continuous affine functions on X. If N is the subspace of A(X)* spanned by the closed face F of X then by making use of the quotient map from A(X)* to A(X)*/N we can extend the definition of "conical" to the closed face F. This is then used to establish a sufficient condition for F to be a peak-face. This procedure of using the quotient map is used repeatedly throughout and as a by-product yields different (and possibly simpler) proofs of some known results. For example we use this approach (see Proposition 4.2) to reprove a result of Alfsen's [2] concerning the complementary face of a closed face of a Choquet simplex. In ?3 we define a class Y of compact convex sets X for which it turns out that (1) every closed G, face F of X is a peak-face and (2) every continuous affine function on F can be extended to a continuous affine function on X. It is known that Choquet simplexes have these two properties and we show that Y in fact contains the simplexes. In addition it is proved that Y contains the a-polytopes. (These are defined by R. Phelps [19] and he proves that they correspond exactly to the polyhedrons defined by Alfsen [1].) In [19] Phelps also defines the 3-polytopes as the intersection of a simplex S with a closed subspace of A(S)* of finite codimension. In ?4 we show that the P-polytopes are conical at each extreme point and that those P-polytopes which are

Introduction. Given a closed additive subset 9)1 of a Banach space, Z, the problem of complementation is fundamental in the study of the structure of the space and also of linear transformations. If we regard e and W2 as abelian groups, it corresponds to the problem of finding in e3 a (closed) subgroup, 9, which is isomorphic to 3/9R under an "element of' relationship. More directly, a complement is defined by the statement: A closed additive set 9 is said to be a complement to 9fl if every element of V3 has a unique resolution in the form g+h where gER and hER. The uniqueness of the resolution is equivalent to the statement: 9)1.91= (0). If we denote the set of elements in the form g+h, gEW9, h 9E, as 9* + *9, complementation can be expressed as + 9 =1, 9 = (O). The existence of a complement is equivalent to the statement: There exists a continuous additive transformation E with E2 = E and range 9fl. Such an E is called a continuous or bounded projection. (Cf. [M2]. The square brackets refer to the bibliography at the end of the paper.) In a previous paper the author has shown that in the separable reflexive Banach space, V,, p>1, p 52, there exist manifolds 9fl which do not have complements ([M2]). A second proof of this has been given in [S]. In an address before the American Mathematical Society on May 2, 1941 ([M4]), the author suggested the study of quasi-complements. 9 is a quasicomplement to WI if WI 9 = (0) and 91fl + * % is dense in Z. For quasi-complements, we have closed rather than continuous projections, E. (Cf. ?4 below.) In the present paper, we prove that for a separable reflexive Banach space Z, with a separable adjoint space, $*, every closed additive set, 9Y, has a quasi-complement. For the type of space mentioned, this answers Problem I of the above mentioned address. In certain ways, this is quite remarkable. For instance, while, in Hilbert space, many essential structure theorems hold for continuous transformations and also for closed transformations (cf. [M3, chap. 91), the latter are preferred simply because they are more inclusive. But here we have a property which is associated with closed projections rather than the continuous projections. In this case, then, the closed transformations are really more effective than the continuous transformations. These results certainly indicate that it will be very desirable to study the structure of linear transformation relative to closed projections as suggested in [M4].