On Sharp Triangle Inequalities in Banach Spaces II

Abstract

Sharp triangle inequality and its reverse in Banach spaces were recently showed by Mitani et al. (2007). In this paper, we present equality attainedness for these inequalities in strictly convex Banach spaces.

1. Introduction

In recent years, the triangle inequality and its reverse inequality have been treated in [1–5] (see also [6, 7]).Kato et al.[8] presented the following sharp triangle inequality and its reverse inequality with elements in a Banach space .

Theorem 1.1 (see [8]).

For all nonzero elements in a Banach space ,

(1.1)

These inequalities are useful to treat geometrical structure of Banach spaces, such as uniform non--ness (see [8]). Moreover,Hsu et al.[9] presented these inequalities for strongly integrable functions with values in a Banach space.

Mitani et al.[10] showed the following inequalities which are sharper than Inequality (1.1) in Theorem 1.1.

Theorem 1.2 (see [10]).

For all nonzero elements in a Banach space with , ,

(1.2)

(1.3)

where .

In this paper we first present a simpler proof of Theorem 1.2. To do this we consider the case as follows.

Theorem 1.3.

For all nonzero elements in a Banach space with

(1.4)

(1.5)

where .

From this result we can easily obtain Theorem 1.2.

Moreover we consider equality attainedness for sharp triangle inequality and its reverse inequality in strictly convex Banach spaces. Namely, we characterize equality attainedness of Inequalities (1.4) and (1.5) in Theorem 1.3.

2. Simple Proofs of Theorems 1.2 and 1.3

Proof of Theorem 1.3.

According to Theorem 1.1 Inequalities (1.4) and (1.5) hold for the case (cf. [3]). Therefore let . We first prove (1.4) by the induction. Assume that (1.4) holds true for all elements in . Let be any elements in with . Let

(2.1)

for all positive numbers with . Then

(2.2)

and . By assumption,

(2.3)

holds, where . Since , from (2.2) and (2.3),

(2.4)

and hence (1.4). Thus (1.4) holds true for all finite elements in .

Next we show Inequality (1.5). Let

(2.5)

Then

(2.6)

and Applying Inequality (1.4) to ,

(2.7)

where . Thus we obtain (1.5). This completes the proof.

Proof of Theorem 1.2.

Let be any nonzero elements in with . For all positive numbers with let

(2.8)

Then . Applying Theorem 1.3 to ,

(2.9)

where for all positive numbers with . As , we have Inequalities (1.4) and (1.5).

3. Equality Attainedness in a Strictly Convex Banach Space

In this section we consider equality attainedness for sharp triangle inequality and its reverse inequality in a strictly convex Banach space. Kato et al. in [8] showed the following.

Theorem 3.1 (see [8]).

Let be a strictly convex Banach space and nonzero elements in . Let and . Let . Then

(3.1)

if and only if either

(a)

or

(b)

Theorem 3.2 (see [8]).

Let be a strictly convex Banach space and nonzero elements in . Let and . Let . Then

(3.2)

if and only if either

(a)

or

(b)

We present equality attainedness for (1.4) and (1.5) in Theorem 1.2. The following lemma given in [8] is quite powerful.

Lemma 3.3 (see [8]).

Let be a strictly convex Banach space. Let be nonzero elements in . Then the following are equivalent.

(i) holds for any positive numbers .

(ii) holds for some positive numbers .

(iii).

Theorem 3.4.

Let be a strictly convex Banach space and nonzero elements in with . Then

(3.3)

if and only if there exists a real number with satisfying .

Proof.

Assume that (3.3) is true. By Theorem 3.1, the equality (3.3) is equivalent to Equality

(3.4)

Put

(3.5)

Then . Since , we obtain . Conversely, if where , then

(3.6)

By , we have (3.4). Thus we get (3.3).

Next we consider the case .

Theorem 3.5.

Let be a strictly convex Banach space and nonzero elements in with . Then

(3.7)

if and only if there exist with satisfying one of the following conditions:

(a)

(b)

Proof.

Assume that (3.7) is true. Put

(3.8)

Then and

(3.9)

Note that . As in the proof of Theorem 1.2 given in [10], we have (3.7) if and only if we have the equalities

(3.10)

(3.11)

By Theorem 3.4, Equality (3.11) implies that

(3.12)

for some with . By (3.8) we have

(3.13)

for some . On the other hand, by Lemma 3.3, Equality (3.10) implies

(3.14)

Hence, by using (3.8), (3.12), and (3.13) we have for some real number . Since , we have . We consider the following two cases

Case 1.

.

Equality (3.14) implies

(3.15)

Hence we have

(3.16)

By and , Equality (3.16) is valid for all real numbers with .

Case 2.

.

Equality (3.14) implies

(3.17)

So we have

(3.18)

Hence . Thus () holds.

Conversely, assume that there exist with satisfying one of the conditions and . Then it is clear that (3.7) holds. Thus we obtain ().

Moreover we consider general cases. For each with , we put For and , we define

(3.19)

For a finite set , the cardinal number of is denoted by .

Lemma 3.6.

Let . If for all with , then

(3.20)

Proof.

Let

(3.21)

where and . By the assumption, we have . We first show for all with . It is clear that . Assume that for all with . We will show . Suppose that . By , we have

(3.22)

Hence we have which is a contradiction. Therefore we have . Namely, for all with . From this result, we obtain . Hence

(3.23)

Theorem 3.7.

Let be a strictly convex Banach space and nonzero elements in with . Then

(3.24)

if and only if there exists with such that

(3.25)

(3.26)

for every with .

Proof.

(): According to Theorems 3.4 and 3.5, Theorem 3.7 is valid for the cases . Therefore let . We will prove Theorem 3.7 by the induction. Assume that this theorem holds true for all nonzero elements in less than . Let and assume that Equality (3.24) holds. Let

(3.27)

for positive integer with . As in the proof of Theorem 1.3, Equality (3.24) holds if and only if

(3.28)

(3.29)

hold, where . Hence, by assumption, there exists with such that

(3.30)

Since by Lemma 3.6, we have

(3.31)

Since

(3.32)

by the definition of , we have

(3.33)

where

(3.34)

Since

(3.35)

we have

(3.36)

for all with . By Lemma 3.3, Equality (3.28) implies

(3.37)

Hence there exists such that . Also,

(3.38)

(3.39)

Since , we have from (3.37) and (3.38),

(3.40)

which implies

(3.41)

If , then it is clear that .

If , then, by (3.41), we have . Hence . Thus we obtain .

(): Let with satisfying (3.25) and (3.26), and let . From (3.25) we have

(3.42)

By Lemma 3.6,

(3.43)

Let , where . From (3.26) and we have

(3.44)

Thus we obtain (3.24). This completes the proof.

In what follows, we characterize the equality condition of Inequality (1.3) in Theorem 1.2. For and positive integer with we define , and

Lemma 3.8.

Let with . If

(3.45)

for all positive integers with , then one has

(3.46)

Proof.

Let and , where and . As in the proof of Lemma 3.6, we have for all . So for all . Hence

(3.47)

This completes the proof.

Theorem 3.9.

Let be a strictly convex Banach space and nonzero elements in with . Then

(3.48)

holds if and only if there exists with , for all positive integers with satisfying

(3.49)

for all positive integers with and .

Proof.

: Let and

(3.50)

For positive integers with we put

(3.51)

Note that and

(3.52)

Then Equality (3.48) holds if and only if

(3.53)

(3.54)

Thus, by the equality condition of sharp triangle inequality with elements, there exists such that

(3.55)

for all positive integers with ,

(3.56)

for all positive integers with . Since

(3.57)

for each with , we have , where

(3.58)

Note that and . By (3.53),

(3.59)

Hence there exists such that . We also have and

(3.60)

Hence we have

(3.61)

From (3.3), we obtain . Thus we have ().

(): Assume that there exists with , for all positive integers with satisfying (3.49) for all positive integers with and . By Theorem 3.7, we have (3.54). From the assumption,

(3.62)

By and Lemma 3.8, we obtain (3.53). Thus we have (). This completes the proof.

If , then we have the following corollary.

Corollary 3.10.

Let be a strictly convex Banach space and nonzero elements in with . Then

(3.63)

if and only if there exists a real number with such that .

If , then we have the following corollary.

Corollary 3.11.

Let be a strictly convex Banach space and nonzero elements in with . Then

(3.64)

if and only if there exist with such that and .

4. Remark

In this section we consider equality attainedness for sharp triangle inequality in a more general case, that is, the case without the assumption that . Let us consider the case .

Proposition 4.1.

Let be a strictly convex Banach space, and nonzero elements in .

(i)If , then the equality

(4.1)

always holds.

(ii)If , then the equality (4.1) holds if and only if there exists a real number satisfying and .

(iii)If , then the equality (4.1) holds if and only if there exists a real number satisfying and .

Proof.

• (i) Is clear.

• (ii) Assume that (4.1) holds. By , (4.1) implies

  

  (4.2)

From Theorem 3.1, we have

(4.3)

Hence for some . The following

(4.4)

implies

(4.5)

Hence and so The converse is clear.

• (iii) Assume that (4.1) holds. Put and . As in the proof of Theorem 3.5, we have

  

  (4.6)

Since , we have for some . The following

(4.7)

implies

(4.8)

Hence . The converse is clear.

Conjecture 1.

What is the necessary and sufficient condition when Equality (3.24) (resp. Equality (3.48)) holds for elements with in Theorem 3.7 (resp. Theorem 3.9)?