TY - JOUR

T1 - Long lines in subsets of large measure in high dimension

AU - Elboim, Dor

AU - Klartag, Bo’az

N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/12

Y1 - 2023/12

N2 - 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 (n1 / 4) . The exponent 1/4 is tight. More generally, for a probability measure μ on Rn and 0 < a< 1 define L(μ,a):=infA;μ(A)=asupℓline|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 ℓp ball. We observe a rather unified behavior in a large class of product measures. On the other hand, for ℓp balls with 1 ≤ p≤ ∞ we find that there are phase transitions of different types.

AB - 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 (n1 / 4) . The exponent 1/4 is tight. More generally, for a probability measure μ on Rn and 0 < a< 1 define L(μ,a):=infA;μ(A)=asupℓline|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 ℓp ball. We observe a rather unified behavior in a large class of product measures. On the other hand, for ℓp balls with 1 ≤ p≤ ∞ we find that there are phase transitions of different types.

KW - High dimension, Radon transform

KW - Needle decomposition

UR - http://www.scopus.com/inward/record.url?scp=85171796920&partnerID=8YFLogxK

U2 - 10.1007/s00440-023-01231-7

DO - 10.1007/s00440-023-01231-7

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AN - SCOPUS:85171796920

SN - 0178-8051

VL - 187

SP - 657

EP - 695

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 3-4

ER -