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- Title
On Extremal Sections of Subspaces of Lp.
- Authors
Eskenazis, Alexandros
- Abstract
Let m , n ∈ N and p ∈ (0 , ∞) . For a finite dimensional quasi-normed space X = (R m , ‖ · ‖ X) , let B p n (X) = (x 1 , ... , x n) ∈ ( R m ) n : ∑ i = 1 n ‖ x i ‖ X p ⩽ 1. We show that for every p ∈ (0 , 2) and X which admits an isometric embedding into L p , the function S n - 1 ∋ θ = (θ 1 , ... , θ n) ⟼ B p n (X) ∩ (x 1 , ... , x n) ∈ ( R m ) n : ∑ i = 1 n θ i x i = 0 is a Schur convex function of (θ 1 2 , ... , θ n 2) , where | · | denotes Lebesgue measure. In particular, it is minimized when θ = (1 n , ... , 1 n ) and maximized when θ = (1 , 0 , ... , 0) . This is a consequence of a more general statement about Laplace transforms of norms of suitable Gaussian random vectors which also implies dual estimates for the mean width of projections of the polar body (B p n (X)) ∘ if the unit ball B X of X is in Lewis' position. Finally, we prove a lower bound for the volume of projections of B ∞ n (X) , where X = (R m , ‖ · ‖ X) is an arbitrary quasi-normed space.
- Subjects
LEBESGUE measure; SCHUR functions; UNIT ball (Mathematics); CONVEX functions; LAPLACE transformation; ISOMETRICS (Mathematics)
- Publication
Discrete & Computational Geometry, 2021, Vol 65, Issue 2, p489
- ISSN
0179-5376
- Publication type
Article
- DOI
10.1007/s00454-019-00133-7