CEU Electronic Theses and Dissertations, 2023
| Author | De, Apratim |
|---|---|
| Title | Some Stability Results in Brunn-Minkowski Theory |
| Summary | This thesis deals with some stability results in Brunn-Minskowski theory, a rich and active field within Convex Geometry and Geometric Functional Analysis. The classical Brunn-Minkowski inequality, the cornerstone of Brunn-Minkowski theory is intimately related to the Minkowski problem and form a core theme of various areas of convex geometry, partial differential equations, probability, additive combinatorics, calculus of variations and others. Firey's and subsequently Lutwak's extension to $L_p$ Minkowski theory naturally gave rise to $L_p$ analogues of the problems, and the geometric and functional inequalities found in the classical theory. Here we focus on obtaining stability results in certain cases for the Prekopa-Leindler inequality (a generalization and functional form of the Brunn-Minkowski inequality), the log-Brunn-Minkowski and log-Minkowski inequalities (which correspond to the $L_0$ versions of the corresponding classical inequalities), and finally, stability of solution of the logarithmic ($L_0$) Minkowski problem. Particularly, we establish: (i) a stability version of the Prekopa-Leindler inequality at least for log-concave functions in $\mathbb{R}^n$ in Chapter 3, (ii) stability versions of the logarithmic Brunn-Minkowski inequality and the Logarithmic Minkowski inequality for convex bodies in $\mathbb{R}^n$ which are symmetric with respect to linear reflections through $n$ independent hyperplanes in Chapter 4, and (iii) stability of solution of the Logarithmic Minkowski Problem on $S^{n-1}$ in the case of symmetries with respect to a Coxeter group $G \subset O(n)$ acting without non-zero fixed points on $\mathbb{R}^n$ in Chapter 5. |
| Supervisor | Böröczky, Károly |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2023/de_apratim.pdf |
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