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.
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 ).
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.
Proof of Theorem 1.2.
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  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 ).
Thus we obtain (3.24). This completes the proof.
This completes the proof.
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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