Local and Global Stability of the - Curvature
Keywords:
curvature function, Minkowski inequality.
Abstract
Origin-centered balls only, when , and only for balls when is the curvature of a smooth, strictly convex body in in known to be constant. Only for origin-symmetric ellipsoids does the --curvature remain constant if . Using the global stability result from [5], we demonstrate that for 0, the volume symmetric difference between K and a translation of the unit ball B is nearly zero if the -curvature is approximately constant. Here, we have K shrunk to the same volume of a unit ball, denoted by K. We demonstrate a comparable result for in the -distance class of origin-symmetric entities. We also demonstrate a local stability conclusion for : Any strictly convex body with 'nearly' constant curvature is 'almost' the unit ball, and this neighborhood surrounds the unit ball. Both a global stability result in R2 for and a local stability result for in the Banach-Mazur distance are demonstrated.
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References
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2. Qazza, A., Abdoon, M., Saadeh, R., & Berir, M. (2023). A New Scheme for Solving a Fractional Differential Equation and a Chaotic System. European Journal of Pure and Applied Mathematics, 16(2), 1128-1139.
3. Saadeh, R., A. Abdoon, M., Qazza, A., & Berir, M. (2023). A Numerical Solution of Generalized Caputo Fractional Initial Value Problems. Fractal and Fractional, 7(4), 332.
4. Abdoon, M. A., Saadeh, R., Berir, M., & Guma, F. E. (2023). Analysis, modeling and simulation of a fractional-order influenza model. Alexandria Engineering Journal, 74, 231-240.
5. Hasan, F. L., & Abdoon, M. A. (2021). The generalized (2+ 1) and (3+ 1)-dimensional with advanced analytical wave solutions via computational applications. International Journal of Nonlinear Analysis and Applications, 12(2), 1213-1241.
6. W.J. Firey, On the shapes of worn stones, Mathematika 21 (1974) 1-11.
7. E. Lutwak, TheBrunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differ. Geom. 38 (1993) 131-150.
8. B. Andrews, Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138 (1999) 151-161.
9. S. Brendle, K. Choi, P. Daskalopoulos, Asymptotic behavior of flows by powers of the Gaussian curvature, Acta Math. 219 (2017) 1-16.
10. Segal, Remark on stability of Brunn-Minkowski and isoperimetric inequalities for convex bodies, in: B. Klartag, S. Mendelson, V.D. Milman (Eds.), Geometric Aspects of Functional Analysis, in: Lecture Notes in Math., vol. 2050, Springer, Berlin, 2012, pp. 381392.
11. K.J. Böröczky, A. De, Stable solution of the log-Minkowski problem in the case of many hyperplane symmetries, J. Differ. Equ. 298 (2021) 298-322.
12. K.J. Böröczky, P. Kalantzopoulos, Log-Brunn-Minkowski inequality under symmetry, Trans. Am. Math. Soc. (2022), https://doi.org/10.1090/tran/8691.
13. S. Chen, H. Yong, Q.R. Li, J. Liu, The -Brunn-Minkowski inequality for , Adv. Math. 368 (2020) 107166.
14. A.V. Kolesnikov, E. Milman, Local -Brunn-Minkowski inequalities for , Mem. Am. Math. Soc. 277 (2022) 1360.
15. E. Milman, Centro-affine differential geometry and the log-Minkowski problem, arXiv:2104.12408, 2021 (35) Volume (3) December 2022; [UBJSR: ISSN [1858-6139]: (Online)
16. E. Milman, A sharp centro-affine isospectral inequality of Szegö-Weinberger type and the -Minkowski problem, J. Differ. Geom. (2022), https://doi.org/10.1007/978-3-64229849-3_24, in press, arXiv:2103.02994.
17. E.C. Gutiérrez, TheMonge-Ampère Equation, vol. 42, Springer Science & Business Media, 2012.
18. S.Y. Cheng, S.T. Yau, Complete affine hypersurfaces. Part I. The completeness of affine metrics, Commun. Pure Appl. Math. 39 (1986) 839-866.
19. K.J. Böröczky, Stability of Blaschke-Santaló inequality and the affine isoperimetric inequality, Adv. Math. 225 (2010) 1914-1928.
20. J.M. Aldaz, A stability version of Hölder's inequality, J. Math. Anal. Appl. 343 (2008) 842-852.
21. E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Mich. Math. J. 5 (1958) 105-126.
22. M. Marini, G. De Philippis, A note on Petty's problem, Kodai Math. J. 37 (2014) 586594.
23. M.N. Ivaki, Deforming a hypersurface by Gauss curvature and support function, J. Funct. Anal. 271 (2016) 2133-2165.
24. L. Simon, Non-linear evolution equations, with applications to geometric problems, Ann. Math.
25. L. Simon, Theorems on Regularity and Singularity of Energy Minimizing Maps, Lectures in Mathematics ETH Zürich, BirkhäuserVerlag, Basel, 1996.
26. Figalli, F. Maggi, F. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math. 182 (2010) 167-211.
27. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, New York, 2014.
28. D. Hug, Curvature relations and affine surface area for a general convex body and its polar, Results Math. 29 (1996) 233-248.
29. M.N. Ivaki, Stability of the Blaschke-Santaló inequality in the plane, Monatshefte Math.
30. M.N. Ivaki, A local uniqueness theorem for minimizers of Petty's conjectured projection inequality, Mathematika 64 (2018) 1-19.
Published
2023-05-15
How to Cite
Salih Yousuf Mohamed Salih, & Shahinaz.A. Elsamani. (2023). Local and Global Stability of the - Curvature. Central Asian Journal of Theoretical and Applied Science, 4(5), 147-162. Retrieved from https://cajotas.centralasianstudies.org/index.php/CAJOTAS/article/view/1176
Section
Articles
