Functional Löwner Ellipsoids

Autor: Grigory Ivanov, Igor Tsiutsiurupa
Rok vydání: 2021
Předmět:
Zdroj: The Journal of Geometric Analysis. 31:11493-11528
ISSN: 1559-002X
1050-6926
DOI: 10.1007/s12220-021-00691-4
Popis: We extend the notion of the minimal volume ellipsoid containing a convex body in $$\mathbb {R}^{d}$$ to the setting of logarithmically concave functions. We consider a vast class of logarithmically concave functions whose superlevel sets are concentric ellipsoids. For a fixed function from this class, we consider the set of all its “affine” positions. For any log-concave function f on $$\mathbb {R}^{d},$$ we consider functions belonging to this set of “affine” positions, and find the one with the minimal integral under the condition that it is pointwise greater than or equal to f. We study the properties of existence and uniqueness of the solution to this problem. For any $$s \in [0,+\infty ),$$ we consider the construction dual to the recently defined John s-function (Ivanov and Naszodi in Functional John ellipsoids. arXiv preprint: arXiv:2006.09934, 2020). We prove that such a construction determines a unique function and call it the Lowner s-function of f. We study the Lowner s-functions as s tends to zero and to infinity. Finally, extending the notion of the outer volume ratio, we define the outer integral ratio of a log-concave function and give an asymptotically tight bound on it.
Databáze: OpenAIRE
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