E extracta mathematicae Vol. 31, Núm. 1, 1 – 10 (2016)

On Small Combination of Slices in Banach Spaces

Sudeshna Basu, T. S. S. R. K. Rao

Department of Mathematics, George Washington University,
Washington DC 20052, USA

sbasu@gwu.edu, sudeshna66@gmail.com

Stat-Math Division, Indian Statistical Institute, R. V. College,
P. O. Bangalore 560059, India

tss@isibang.ac.in, srin@fulbrightmail.org

Presented by Pier L. Papini Received August 13, 2015

Abstract: The notion of Small Combination of Slices (SCS) in the unit ball of a Banach
space was first introduced in [4] and subsequently analyzed in detail in [12] and [13]. In this
work, we introduce the notion of BSCSP, which can be seen as a generalization of dentability
in terms of SCS. We study certain stability results for the w∗-BSCSP leading to a discussion
on BSCSP in the context of ideals of Banach spaces. We prove that the w∗-BSCSP can be
lifted from a M-ideal to the whole Banach Space. We also prove similar results for strict
ideals and U-subspaces of a Banach space. We note that the space C(K, X)∗ has w∗-BSCSP
when K is dispersed and X∗ has the w∗-BSCSP.

Key words: Small combination of slices, M-Ideals, Strict ideals, U-Subspaces.

AMS Subject Class. (2010): 46B20, 46B28.

1. Introduction

Let X be a real Banach space and X∗ its dual. We will denote by BX, SX
and BX(x, r) the closed unit ball, the unit sphere and the closed ball of radius
r > 0 and center x. We refer to the monograph [2] for notions of convexity
theory that we will be using here.

Definition 1. (i) We say A ⊆ BX∗ is a norming set for X if ∥x∥ =
sup{|x∗(x)| : x∗ ∈ A}, for all x ∈ X. A closed subspace F ⊆ X∗ is a norming
subspace if BF is a norming set for X.

(ii) Let f ∈ X∗, α > 0 and C ⊆ X. Then the set

S(C, f, α) = {x ∈ C : f(x) > sup f(C) − α}

is called the open slice determined by f and α. We assume without loss of
generality that ∥f∥ = 1. One can analogously define w∗ slices in X∗

1



2 s. basu, t.s.s.r.k. rao

(iii) A point x ̸= 0 in a convex set K ⊆ X is called a SCS (small combination
of slices) point of K, if for every ε > 0, there exist slices Si of K, and a
convex combination S =

∑n
i=1 λiSi such that x ∈ S and diam(S) < ε. One

can analogously define w∗-SCS point in X∗.

We introduce the following definition analogous to that of a unit ball being
dentable, see [2].

Definition 2. A Banach Space is said to have Ball-Small Combination
of Slices Property (BSCSP) if the unit ball has small combination of slices of
arbitrarily small diameter. Analogously we can define w∗-BSCSP in a dual
space.

Remark 3. (i) It is clear that if BX has a SCS point, then it has BSCSP.

(ii) Strongly Regular spaces studied in [4] and [13] were referred to as Small
Combination of Slices Property (SCSP) in [12].

SCS points were first introduced in [4] as a “slice generalization” of the
notion PC (i.e. points for which the identity mapping on the unit ball, from
weak topology to norm topology is continuous). It was proved in [4] that X is
strongly regular (respectively, X∗ is w∗-strongly regular) if and only if every
non empty bounded convex set K in X (respectively K in X∗) is contained
in the norm closure (respectively, w∗-closure) of SCS(K) (respectively w∗-
SCS(K)), i.e. the SCS points (w∗-SCS points) of K. Later, it was proved in
[13] that Banach space has Radon Nikodym Property (RNP) if and only if it
is strongly regular and has the Krein-Milman Property (KMP). Subsequently,
the concept of SCS points was used in [12] to investigate the structure of non
dentable closed bounded convex sets in Banach spaces. In this work, we study
certain stability results for w∗−BSCSP leading to a discussion on BSCSP
in the context of ideals of Banach spaces, see [5] and [12]. We use various
techniques from the geometric theory of Banach spaces to achieve this. The
spaces that we will be considering have been well studied in the literature.
A large class of function spaces like the Bloch spaces, Lorentz and Orlicz
spaces, spaces of vector-valued functions and spaces of compact operators are
examples of the spaces we will be considering: for details, see [6]. We provide
some descriptions of w∗-SCS points in Banach spaces in different contexts.
We need the following definition.



on small combination of slices in banach spaces 3

Definition 4. Let X be a Banach space.

(i) A linear projection P on X is called an M-projection if

∥x∥ = max{∥Px∥, ∥x − Px∥},

for all x ∈ X; A linear projection P on X is called an L-projection if

∥x∥ = ∥Px∥ + ∥x − Px∥

for all x ∈ X.
(ii) A subspace M ⊆ X is called an M-summand if it is the range of an M-
projection. A closed subspace M ⊆ X is called an L-summand if it is the
range of an L-projection.

(iii) A subspace M ⊆ X is called an M-ideal if M⊥ is the kernel of an L-
projection in X∗

We recall from [6, Chapter I] that when M ⊂ X is an M-ideal, elements
of M∗ have unique norm-preserving extension to X∗ and one has the identifi-
cation, X∗ = M∗ ⊕1 M⊥. Several examples from among function spaces and
spaces of operators that satisfy these geometric properties can be found in the
monograph [6], see also [8]. First, we prove that for an L-summand M ⊂ X, if
a SCS point of BX has a non-zero component m ∈ M, then m is a SCS point
of BM. For an M- ideal M ⊂ X, this yields: any w∗-SCS point of BX∗, if its
restriction to M, say m∗, has the same norm, then m∗ it is a w∗-SCS point
of BM∗ . We prove a similar result for a U-subspace of a Banach space of X.
We prove a converse statement for a strict ideal Y ⊂ X (see Section 2 for the
definition) i.e., we prove that a w∗-SCS point of a strict ideal continues to be
so in the bigger space. We also prove corresponding results for the BSCSP.

2. Stability results

We will use the standard notation of ⊕1, ⊕∞ to denote the ℓ1 and ℓ∞-direct
sum of two or more Banach spaces.

Proposition 5. Suppose X, Y , Z are Banach spaces such that Z =
X ⊕1 Y ; suppose z0 = (x0, y0) ∈ BZ is a SCS point of BZ with both the
components non-zero, then x0 and y0 are SCS points of BX and BY respec-
tively.



4 s. basu, t.s.s.r.k. rao

Proof. Since z0 is a SCS point of BZ, we have for any ε > 0, z0 =
∑n

i=1 λizi,
where zi ∈ Si and for z∗i = (x

∗
i , y

∗
i ) with 1 = ∥z

∗
i ∥ = max{∥x

∗
i ∥, ∥y

∗
i ∥},

Si = {z ∈ BZ/z∗i (z) > 1 − εi} and diam(
∑n

i=1 λiSi) < ε,

Si = {z ∈ BZ/z∗i (x, y) > 1 − εi} = {z ∈ BZ/x
∗
i (x) + y

∗
i (y) > 1 − εi}.

Since zi = (xi, yi) ∈ Si, then x∗i (xi) + y
∗
i (yi) > 1 − εi.

Case 1 : ∥z∗i ∥ = ∥x
∗
i ∥ = 1. Then,

x∗i (xi) + y
∗
i (yi) > 1 − εi = ∥x

∗
i ∥ − εi,

=⇒ x∗i (xi) > ∥x
∗
i ∥ − εi − y

∗
i (yi),

=⇒ 1 ≥ x∗i (xi) > ∥x
∗
i ∥ − βi, where βi = εi + y

∗
i (yi),

=⇒ εi + y∗i (yi) > 0.

So we have, xi ∈ SiX ={x ∈BX/x∗i (x)>1−βi}. Then (xi, yi)∈ SiX×{yi}⊆Si.
Case 2: ∥z∗i ∥ = ∥y

∗
i ∥ = 1. We may assume that 0 < ∥x

∗
i ∥ < 1, and let

δi = ∥y∗i ∥ − ∥x
∗
i ∥. Then,

x∗i (xi) + y
∗
i (yi) > 1 − εi = ∥y

∗
i ∥ − εi = ∥x

∗
i ∥ + δi − εi

=⇒ x∗i (xi) > ∥x
∗
i ∥ + δi − εi − y

∗
i (yi),

=⇒ ∥x∗i ∥ ≥ x
∗
i (xi) > ∥x

∗
i ∥ − ri, where ri = δi − εi − y

∗
i (yi) > 0,

=⇒ xi ∈ SiX = {x ∈ BX/x∗i (x) > 1 − ri}.

Then (xi, yi) ∈ SiX × {yi} ⊆ Si.

Let x0 =
∑n

i=1 λixi and y0 =
∑n

i=1 λiyi. Now x0 ∈
∑n

i=1 λiSiX. Also,

n∑
i=1

λi[SiX × yi] ⊆
n∑

i=1

λiSi,

=⇒
n∑

i=1

λi[SiX] × {y0} ⊆
n∑

i=1

λi[SiX × yi] ⊆
n∑

i=1

λiSi,

=⇒ diam
( n∑

i=1

λiSiX

)
< ε,

=⇒ x0 is a SCS point of BX.

Similarly it follows that y0 is a SCS point of BY .



on small combination of slices in banach spaces 5

Arguments similar to the ones given above in the context of a ℓ∞-sum
yield the following corollary.

Corollary 6. Suppose X, Y , Z are Banach spaces such that Z =
X ⊕∞ Y , suppose z∗ = (x∗, y∗) ∈ BZ∗ is a w∗-SCS point of BZ∗ with both
the components non-zero, then x∗ and y∗ are w∗-SCS points of BX∗ and BY ∗

respectively.

Remark 7. Since in the sequence space ℓ∞ any weakly open set has norm
diameter 2, by taking X = c0 and Y = ℓ

1, Z = X ⊕∞ Y , any w∗-SCS point of
BZ∗ has its second component 0. We thank the referee for this observation.

Definition 8. We recall that a closed subspace Y of a Banach space X
is called a U-subspace if for y∗ ∈ Y ∗ there exists a unique norm preserving
extension of y∗ in X∗. We continue to denote the unique extension also by y∗.

See the discussion on [6, page 44] and the references in that monograph
for several examples of U-subspaces from among classical function spaces and
spaces of operators.

Before the next result we also need a definition from [5]. See also [11] for
more information and several examples from spaces of operators and tensor
product spaces.

Definition 9. A closed subspace Y of a Banach Space X is said to be an
ideal of X if there is a linear projection P : X∗ → X∗ of norm one such that
ker(P) = Y ⊥.

For x∗ ∈ X∗ since P(x∗) − x∗ = 0 on Y , as ∥P∥ = 1, we see that P(x∗) is
a norm-preserving extension of x∗|Y .

Theorem 10. Suppose Y is an ideal which is also a U-subspace of X. If
y∗ ∈ SY ∗ is a w∗-SCS point of BX∗, then y∗ is a w∗-SCS point of BY ∗.

Proof. Let y∗0 ∈ SY ∗ be a w
∗-SCS point of BX∗ , hence for any ε > 0 there

exist w∗ slices Si, 0 ≤ λi ≤ 1, i = 1, 2, . . . , n, Si = {x∗ ∈ BX∗/x∗(xi) > 1−αi}
and diam(

∑n
i=1 λiSi) < ε and y

∗
0 =

∑
λix

∗
0i. Since y

∗
0 ∈ SY ∗ and Y is a U-

subspace, y∗0 has unique norm preserving extension in X
∗. Let P : X∗ −→ X∗

be the canonical projection. Then ∥P(y∗0)∥ = ∥y
∗
0∥ = 1, Also,

1 = ∥y∗0∥ =
∥∥∥∥ n∑

i=1

λix
∗
0i

∥∥∥∥ ≤ n∑
i=1

λi∥P(x
∗
0i)∥ ≤ 1.



6 s. basu, t.s.s.r.k. rao

This implies ∥P(x∗0i)∥ = ∥x0i
∗∥ = 1 for all i = 1, . . . , n. Thus by hypothesis,

P(x∗0i) and the restriction of x
∗
0i to Y are denoted by y

∗
0i. Now y

∗
0i ∈ Si,

then y∗0i(xi) > 1 − αi. Also, since Y is an ideal, there exists an operator
T : span{xi} −→ Y such that ∥T(xi)∥ ≤ (1 + ε)∥xi∥ = 1 + ε.

Let yi = T(xi). Hence,

y∗0i(xi) > 1 − αi =⇒ y
∗
0i(yi − yi + xi) > 1 − αi,

=⇒ y∗0i(yi) + y
∗
0i(xi − yi) > 1 − αi,

=⇒ y∗0i(yi) > 1 − αi − y
∗
0i(xi − yi).

Case 1: ∥yi∥ =1. So we have

1 > y∗0i(yi) > 1 − αi−y
∗
0i(xi − yi) = 1 − βi,

=⇒ y∗0i ∈ SiY = {y
∗ ∈ BY ∗/y∗(yi) > 1 − βi}.

Case 2: ∥yi∥ < 1. Let ∥yi∥ = 1 − δi. Then

∥yi∥ > y∗0i(yi) > ∥yi∥ + δi − βi = ∥yi∥ − (βi − δi) = ∥yi∥ − γi, γi > 0,

=⇒ y∗0i ∈ SiY = {y
∗ ∈ BY ∗/y∗(yi) > ∥yi∥ − γi}.

Case 3: ∥yi∥ = 1 + δi. Then

1 + δi > y
∗
0i(yi) > 1 − βi = 1 + δi − (βi + δi),

=⇒ y∗0i ∈ SiY = {y
∗ ∈ BY ∗/y∗(yi) > ∥yi∥ − (βi + δi}.

Hence

y∗0 =
n∑

i=1

λiy
∗
0i ∈

n∑
i=1

λiSiY ⊆
n∑

i=1

λiSi.

Hence

diam

( n∑
i=1

λiSiY

)
< diam

( n∑
i=1

λiSi

)
< ε.

Thus y∗0 is w
∗-SCS point of BY ∗.

Let M ⊆ X be an M-ideal. It follows from the results in [6, Chapter I]
that any x∗ ∈ X∗, if ∥m∗∥ = ∥x∗|M∥ = ∥x∗∥, then x∗ is the unique norm
preserving extension of m∗. For notational convenience we denote both the
functionals by m∗. Clearly any M-ideal is also an ideal. Thus the following
corollary answers a natural question in this context for w∗-SCS points of the
unit sphere. We omit its easy proof.



on small combination of slices in banach spaces 7

Corollary 11. Suppose M ⊆ X is a M-ideal in X. If m∗ ∈ SX∗ is
w∗-SCS point of BX∗, then m

∗ ∈ SM∗ is a w∗-SCS point of BM∗.

Remark 12. The referee has kindly pointed out an independent proof to
show that for Z = X ⊕1 Y , Z has the BSCSP if and only if X or Y has the
BSCSP.

Arguments similar to the ones given during the proof of Proposition 5 can
be used to show that for Z = X ⊕∞ Y , if X∗ or Y ∗ has the w∗-BSCSP then
so does Z∗.

In the case of an M-ideal M ⊂ X, for the sake of completeness we give a
detailed proof of the following result.

Proposition 13. Let M ⊆ X be a M-ideal, then if M∗ has the w∗-
BSCSP then X∗ has the w∗-BSCSP.

Proof. Suppose M∗ has the w∗-BSCSP, then for any ε > 0 there exists
slices SiM and 0 ≤ λi ≤ 1, i = 1, 2, . . . , n, SiM = {m∗ ∈ BM∗/m∗(mi) >
1 − αi} and diam(

∑n
i=1 λiSiM) < ε. Since M is an M- ideal, for any x

∗ ∈ X∗
we have the unique decomposition, x∗ = m∗ + m⊥, where m∗ ∈ M∗ and
m⊥ ∈ M⊥. Suppose we have 0 < µi < αi. Then

SiX = {x∗ ∈ BX∗/x∗(mi) > 1 − µi}
= {x∗ ∈ BX∗/m∗(mi) + m⊥(mi) > 1 − µi},
⊆ SiM × µiBM⊥,

=⇒
n∑

i=1

λiSiX ⊆
n∑

i=1

λiSiM × µiBM⊥.

Choose βi = min(µi, ε). Then

S′iX = {x
∗ ∈ BX∗/x∗(mi) > 1 − βi} ⊆ SiX × βiBM⊥,

=⇒
n∑

i=1

λiS
′
iX ⊆

( n∑
i=1

λiSiM × βiBM⊥
)

=⇒
n∑

i=1

λiS
′
iX ⊆

( n∑
i=1

λiSiM × βiBM⊥
)
.

Thus diam(
∑n

i=1 λiS
′
iX) ≤ diam(

∑n
i=1 λiSiM) + 2ε < ε + 2ε = 3ε. Also, since

∥mi∥ = 1, there exists m∗i ∈ BM∗ such that m
∗
i (mi) > 1−βi. Hence m

∗
i ∈ S

′
iX.

Similarly,
∑n

i=1 λim
∗
i ∈

∑n
i=1 λiS

′
iX =⇒

∑n
i=1 λiS

′
i ̸= ∅.



8 s. basu, t.s.s.r.k. rao

Since any summand in a ℓ∞-direct sum is in particular an M-ideal of the
sum, the following corollary is easy to prove.

Corollary 14. Suppose X = ⊕ℓ∞Xi. If X∗i has the w
∗-BSCSP for some

i, then X∗ has the w∗-BSCSP.

The above arguments extend easily to vector-valued continuous functions.
We recall that for a compact Hausdorff space K, C(K, X) denotes the space of
continuous X-valued functions on K, equipped with the supremum norm. We
recall from [9] that dispersed compact Hausdorff spaces have isolated points.

Corollary 15. Suppose K is a compact Hausdorff space with an isolated
point. If X∗ has the w∗-BSCSP, then C(K, X)∗ has the w∗-BSCSP.

Proof. Suppose X∗ has the w∗-BSCSP. For an isolated point k0 ∈ K, the
map F → χk0F is an M-projection in C(K, X) whose range is isometric to
X. Hence we see that C(K, X)∗ has the w∗-BSCSP.

We recall that an ideal Y is said to be a strict ideal if for a projection
P : X∗ → X∗ with ∥P∥ = 1, ker(P) = Y ⊥, one also has BP(X∗) is w∗-dense
in BX∗ or in other words BP(X∗) is a norming set for X.

In the case of an ideal also one has that Y ∗ embeds (though there may not
be uniqueness of norm-preserving extensions) as P(X∗). Thus we continue to
write X∗ = Y ∗ ⊕Y ⊥. In what follows we use a result from [11], that identifies
strict ideals as those for which Y ⊂ X ⊂ Y ∗∗ under the canonical embedding
of Y in Y ∗∗.

Proposition 16. Suppose Y is a strict ideal of X. If y∗ ∈ BY ∗ is a
w∗-SCS point of BY ∗, then y

∗ is a w∗-SCS point of BX∗.

Proof. Since y∗ ∈ BY ∗ is a w∗-SCS point of BY ∗, for any ε > 0 there exists
w∗ slices Si and 0 ≤ λi ≤ 1, i = 1, 2, . . . , n, Si = {y∗ ∈ BY ∗/y∗(yi) > 1 − αi}
and diam(

∑n
i=1 λiSi) < ε. Since Y is a strict ideal in X, we have BX∗ =

BY ∗
w∗

, hence we have the following:

S′i = {x
∗ ∈ BX∗/x∗(xi) > 1 − αi} = {x∗ ∈ BY ∗

w∗
/x∗(xi) > 1 − αi},

=⇒ diam
( n∑

i=1

λiS
′
i

)
⊆ diam

( n∑
i=1

λiSi

)
< ε,

=⇒ diam
( n∑

i=1

λiS
′
i

)
< ε.



on small combination of slices in banach spaces 9

Hence y∗ is a w∗-SCS point of BY ∗.

Arguing similarly it follows that:

Proposition 17. Suppose Y is a strict ideal of X. If Y ∗ has w∗-BSCSP
then X∗ has w∗-BSCSP.

Remark 18. A prime example of a strict ideal is a Banach space X under
its canonical embedding in X∗∗. It is known that any w∗-denting point of BX∗∗

is a point of X. Now let x∗∗ ∈ BX∗∗ be a w∗-SCS point. The referee has kindly
pointed out that since BX is weak

∗ dense in BX∗∗, for any ϵ > 0, there is a
convex combination

∑n
i=1 λixi of vectors in X so that ∥x

∗∗ −
∑n

i=1 λixi∥ ≤ ϵ.
Hence x∗∗ ∈ X.

We conclude the paper with a set of remarks and questions. See also the
recent paper [1] for other possible geometric connections. Let us consider the
following densities of w∗-SCS points of BX∗.

(i) All points of SX∗ are w
∗-SCS points of BX∗.

(ii) The w∗-SCS points of BX∗ are dense in SX∗.

(iii) BX∗ is contained in the closure of w
∗-SCS points of BX∗.

(iv) BX∗ is the closed convex hull of w
∗-SCS points of BX∗.

(v) X∗ is the closed linear span of w∗-SCS points of BX∗.

Questions:

(i) How can each of these properties be realized as a ball separation property
considered in [3]?

(ii) What stability results will hold for these properties?

Acknowledgements

This work was done when the first author was visiting Indian Sta-
tistical Institute Bangalore Center. She would like to express her deep
gratitude to Professor T. S. S. R. K. Rao and everyone at ISI Bangalore
Center, for the warm hospitality provided during her stay. The second
author currently is a Fulbright-Nehru Academic and Professional Excel-
lence Fellow, at the Department of Mathematical Sciences, University
of Memphis. The authors would also like to thank Professor Pradipta
Bandyopadhyay for some discussions they had with him. The authors
thank the referee for pointing out inaccuracies in earlier versions and
other comments for reorganizing and improving the presentation.



10 s. basu, t.s.s.r.k. rao

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