We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Long lines in subsets of large measure in high dimension.
- Authors
Elboim, Dor; Klartag, Bo'az
- Abstract
We show that for any set A ⊆ [ 0 , 1 ] n with Vol (A) ≥ 1 / 2 there exists a line ℓ such that the one-dimensional Lebesgue measure of ℓ ∩ A is at least Ω (n 1 / 4) . The exponent 1/4 is tight. More generally, for a probability measure μ on R n and 0 < a < 1 define L (μ , a) : = inf A ; μ (A) = a sup ℓ line | ℓ ∩ A | where | · | stands for the one-dimensional Lebesgue measure. We study the asymptotic behavior of L (μ , a) when μ is a product measure and when μ is the uniform measure on the ℓ p ball. We observe a rather unified behavior in a large class of product measures. On the other hand, for ℓ p balls with 1 ≤ p ≤ ∞ we find that there are phase transitions of different types.
- Subjects
LEBESGUE measure; PHASE transitions; RADON transforms; PROBABILITY measures
- Publication
Probability Theory & Related Fields, 2023, Vol 187, Issue 3/4, p657
- ISSN
0178-8051
- Publication type
Article
- DOI
10.1007/s00440-023-01231-7