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Improved estimates for the triangle inequality
Nicuşor Minculete1, Radu Păltănea1
1Faculty of Mathematics and Computer Science, Transilvania University of Braşov, Str. Iuliu Maniu, nr. 50, Braşov, 500091 Romania.
Journal of Inequalities and Applications
|January 31, 2017
Summary
This study refines estimates of the triangle inequality in normed spaces using integration and the Tapia semi-product. These new findings offer a more detailed understanding, particularly for inner product spaces.
Area of Science:
- Mathematics
- Functional Analysis
Background:
- The triangle inequality is a fundamental concept in metric spaces, including normed and inner product spaces.
- Existing bounds for the triangle inequality have been explored, but refined estimates are continually sought.
Purpose of the Study:
- To derive improved quantitative estimates for the triangle inequality in the context of normed spaces.
- To investigate the application of integral methods and the Tapia semi-product for refining these estimates.
- To specifically analyze the implications for inner product spaces.
Main Methods:
- Utilizing integral calculus to establish new bounds.
- Employing the Tapia semi-product, a generalization of inner products, within the analysis.
- Comparing and contrasting results in general normed spaces with the specific case of inner product spaces.
Main Results:
- Obtained refined estimates for the triangle inequality in normed spaces.
- Demonstrated the effectiveness of integral techniques and the Tapia semi-product in achieving these refined bounds.
- Provided a more detailed analysis and specific results for inner product spaces.
Conclusions:
- The study successfully provides refined estimates for the triangle inequality, enhancing theoretical understanding.
- The methods employed offer a powerful approach for further investigations in geometric analysis.
- The detailed examination of inner product spaces highlights the practical applicability of the developed techniques.
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