# On Weak Parallelogram Constants for Lebesgue Spaces

### Abstract

This paper comprehensively examines fundamental principles pertaining to parallelogram and parallelogram (abbreviated as w.p.) laws. Emphasis is placed on investigating the duality in the w.p. laws, alongside a thorough review and analysis of foundational findings concerning the geometry of w.p. spaces. Additionally, the research delves into the exploration of optimal w.p. constants for Lebesgue spaces. Through rigorous examination and synthesis of pertinent literature, this study aims to provide a comprehensive understanding of the theoretical underpinnings and practical implications of parallelogram laws and their application in Lebesgue spaces.

### References

D. C. Kay, “A Parallelogram Law for Certain Lp Spaces,” The American Mathematical Monthly, 1967, doi: 10.1080/00029890.1967.11999932.

W. L. Bynum, “Weak parallelogram laws for Banach spaces,” Canadian Mathematical Bulletin, 1976, [Online]. Available: https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/weak-parallelogram-laws-for-banach-spaces/1BCA3CA7148F793378A64D85CB9D4EBF

Z. B. Xu and G. F. Roach, “Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces,” J Math Anal Appl, 1991, [Online]. Available: https://www.sciencedirect.com/science/article/pii/0022247X9190144O

R. Cheng, A. G. Miamee, and M. Pourahmadi, “On the geometry of Lp (μ) with applications to infinite variance processes,” Journal of the Australian …, 2003, [Online]. Available: https://www.cambridge.org/core/journals/journal-of-the-australian-mathematical-society/article/on-the-geometry-of-lp-with-applications-to-infinite-variance-processes/12EB97D22705D3FBA760B460C89F1A66

R. Cheng and C. B. Harris, “Duality of the weak parallelogram laws on Banach spaces,” J Math Anal Appl, 2013, [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0022247X13001790

R. Cheng and W. T. Ross, “Weak parallelogram laws on Banach spaces and applications to prediction,” Period Math Hung, 2015, doi: 10.1007/s10998-014-0078-4.

J. J. Sopka, “Introductory Functional Analysis with Applications (Erwin Kreyszig),” SIAM Review, vol. 21, no. 3, pp. 412–413, 1979, doi: 10.1137/1021075.

A. H. Siddiqi and S. Nanda, Functional analysis and applications. Springer, 2018.

R. G. Bartle, The elements of integration and Lebesgue measure. John Wiley & Sons, 2014.

W. Rudin, Principles of mathematical analysis, vol. 3. McGraw-hill New York, 1976.

G. B. Folland, Real analysis: modern techniques and their applications, vol. 40. John Wiley & Sons, 1999.

P. Jordan and J. V. Neumann, “On Inner Products in Linear, Metric Spaces,” The Annals of Mathematics, vol. 36, no. 3, p. 719, Jul. 1935, doi: 10.2307/1968653.

M. M. Day, “Normed linear spaces, „Academic Press,” Inc., New York, 1962.

C. E. Chidume, “Iterative methods and nonlinear functional equations,” Ohio State University, 1984.

N. L. Carothers, A short course on Banach space theory, no. 64. Cambridge University Press, 2005.

W. L. Bynum and J. H. Drew, “A weak parallelogram law for lp,” The American Mathematical Monthly, vol. 79, no. 9, pp. 1012–1015, 1972.

J. Tian, “An experimental investigation of the fracturing behaviour of rock-like materials containing two V-shaped parallelogram flaws,” Int J Min Sci Technol, vol. 30, no. 6, pp. 777–783, 2020, doi: 10.1016/j.ijmst.2020.07.002.

B. B. Mohsen, “Strongly convex functions of higher order involving bifunction,” Mathematics, vol. 7, no. 11, 2019, doi: 10.3390/math7111028.

G. Jia, “Type synthesis of plane-symmetric deployable grasping parallel mechanisms using constraint force parallelogram law,” Mech Mach Theory, vol. 161, 2021, doi: 10.1016/j.mechmachtheory.2021.104330.

M. A. Noor, “Higher order strongly general convex functions and variational inequalities,” AIMS Mathematics, vol. 5, no. 4, pp. 3646–3663, 2020, doi: 10.3934/math.2020236.

X. Li, “Study on Law of Overlying Strata Breakage and Migration in Downward Mining of Extremely Close Coal Seams by Physical Similarity Simulation,” Advances in Civil Engineering, vol. 2020, 2020, doi: 10.1155/2020/2898971.

M. A. Noor, “Higher-order strongly-generalized convex functions,” Applied Mathematics and Information Sciences, vol. 14, no. 1, pp. 133–139, 2020, doi: 10.18576/AMIS/140117.

L. Huang, “Atomic and Littlewood–Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications,” J Geom Anal, vol. 29, no. 3, pp. 1991–2067, 2019, doi: 10.1007/s12220-018-0070-y.

A. Durmus, “High-dimensional Bayesian inference via the unadjusted Langevin algorithm,” Bernoulli, vol. 25, no. 4, pp. 2854–2882, 2019, doi: 10.3150/18-BEJ1073.

I. Bisio, “Variable-Exponent Lebesgue-Space Inversion for Brain Stroke Microwave Imaging,” IEEE Trans Microw Theory Tech, vol. 68, no. 5, pp. 1882–1895, 2020, doi: 10.1109/TMTT.2019.2963870.

T. Nogayama, “Mixed Morrey spaces,” Positivity (Dordr), vol. 23, no. 4, pp. 961–1000, 2019, doi: 10.1007/s11117-019-00646-8.

A. Fedeli, “Nonlinear S-Parameters Inversion for Stroke Imaging,” IEEE Trans Microw Theory Tech, vol. 69, no. 3, pp. 1760–1771, 2021, doi: 10.1109/TMTT.2020.3040483.

Y. Zhang, “Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: Decompositions with applications to boundedness of Calderón-Zygmund operators,” Sci China Math, vol. 64, no. 9, pp. 2007–2064, 2021, doi: 10.1007/s11425-019-1645-1.

Y. Sawano, “A THOUGHT ON GENERALIZED MORREY SPACES,” Journal of the Indonesian Mathematical Society, vol. 25, no. 3, pp. 210–281, 2019, doi: 10.22342/JIMS.25.3.819.210-281.

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*5*(2), 36-51. Retrieved from https://cajmtcs.centralasianstudies.org/index.php/CAJMTCS/article/view/616