Functional Löwner Ellipsoids

نویسندگان

چکیده

We extend the notion of minimal volume ellipsoid containing a convex body in $$\mathbb {R}^{d}$$ to setting logarithmically concave functions. consider vast class functions whose superlevel sets are concentric ellipsoids. For fixed function from this class, we set all its “affine” positions. any log-concave f on {R}^{d},$$ belonging positions, and find one with integral under condition that it is pointwise greater than or equal f. study properties existence uniqueness solution problem. $$s \in [0,+\infty ),$$ construction dual recently defined John s-function (Ivanov Naszódi Functional arXiv preprint: arXiv:2006.09934, 2020). prove such determines unique call Löwner s-functions as s tends zero infinity. Finally, extending outer ratio, define ratio give an asymptotically tight bound it.

برای دانلود باید عضویت طلایی داشته باشید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

A generalization of Löwner-John's ellipsoid theorem

We address the following generalization P of the LöwnerJohn ellipsoid problem. Given a (non necessarily convex) compact set K ⊂ R and an even integer d ∈ N, find an homogeneous polynomial g of degree d such that K ⊂ G := {x : g(x) ≤ 1} and G has minimum volume among all such sets. We show that P is a convex optimization problem even if neither K nor G are convex! We next show that P has a uniqu...

متن کامل

Löwner ’ s equation in noncommutative probability ∗

Using concepts of noncommutative probability we show that the Löwner’s evolution equation can be viewed as providing a map from paths of measures to paths of probability measures. We show that the fixed point of the Löwner map is the convolution semigroup of the semicircle law in the chordal case, and its multiplicative analogue in the radial case. We further show that the Löwner evolution “spr...

متن کامل

Löwner-john Ellipsoids

Before giving the mathematical description of the Löwner-John ellipsoids and pointing out some of their far-ranging applications, I briefly illuminate the adventurous life of the two eminent mathematicians, by whom the ellipsoids are named: Charles Loewner (Karel Löwner) and Fritz John. Karel Löwner was born into a Jewish family in Lány, a small town about 30 km west of Prague, in 1893. Due to ...

متن کامل

Packing Ellipsoids with Overlap

The problem of packing ellipsoids of different sizes and shapes into an ellipsoidal container so as to minimize a measure of overlap between ellipsoids is considered. A bilevel optimization formulation is given, together with an algorithm for the general case and a simpler algorithm for the special case in which all ellipsoids are in fact spheres. Convergence results are proved and computationa...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

ژورنال

عنوان ژورنال: Journal of Geometric Analysis

سال: 2021

ISSN: ['1559-002X', '1050-6926']

DOI: https://doi.org/10.1007/s12220-021-00691-4