- Research Article
- Open Access
On Sharp Triangle Inequalities in Banach Spaces II
© K.-I. Mitani and K.-S. Saito. 2010
- Received: 4 November 2009
- Accepted: 2 March 2010
- Published: 28 April 2010
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.
- Banach Space
- Positive Integer
- Equality Condition
- Integrable Function
In recent years, the triangle inequality and its reverse inequality have been treated in [1–5] (see also [6, 7]).Kato et al. presented the following sharp triangle inequality and its reverse inequality with elements in a Banach space .
Theorem 1.1 (see ).
These inequalities are useful to treat geometrical structure of Banach spaces, such as uniform non- -ness (see ). Moreover,Hsu et al. presented these inequalities for strongly integrable functions with values in a Banach space.
Mitani et al. showed the following inequalities which are sharper than Inequality (1.1) in Theorem 1.1.
Theorem 1.2 (see ).
In this paper we first present a simpler proof of Theorem 1.2. To do this we consider the case as follows.
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.
Proof of Theorem 1.3.
and hence (1.4). Thus (1.4) holds true for all finite elements in .
where . Thus we obtain (1.5). This completes the proof.
Proof of Theorem 1.2.
where for all positive numbers with . As , we have Inequalities (1.4) and (1.5).
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  showed the following.
Theorem 3.1 (see ).
if and only if either
Theorem 3.2 (see ).
if and only if either
We present equality attainedness for (1.4) and (1.5) in Theorem 1.2. The following lemma given in  is quite powerful.
Lemma 3.3 (see ).
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 .
if and only if there exists a real number with satisfying .
By , we have (3.4). Thus we get (3.3).
Next we consider the case .
if and only if there exist with satisfying one of the following conditions:
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
By and , Equality (3.16) is valid for all real numbers with .
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 ( ).
For a finite set , the cardinal number of is denoted by .
for every with .
If , then it is clear that .
If , then, by (3.41), we have . Hence . Thus we obtain .
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
This completes the proof.
for all positive integers with and .
From (3.3), we obtain . Thus we have ( ).
By and Lemma 3.8, we obtain (3.53). Thus we have ( ). This completes the proof.
If , then we have the following corollary.
if and only if there exists a real number with such that .
If , then we have the following corollary.
if and only if there exist with such that and .
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 .
Let be a strictly convex Banach space, and nonzero elements in .
(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 .
Hence . The converse is clear.
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)?
The second author was supported in part by Grants-in-Aid for Scientific Research (no. 20540158), Japan Society for the Promotion of Science.
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