Long lines in subsets of large measure in high dimension

Dor Elboim, Bo’az Klartag*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)657-695
Number of pages39
JournalProbability Theory and Related Fields
Issue number3-4
StateAccepted/In press - 2023
Externally publishedYes


  • High dimension, Radon transform
  • Needle decomposition


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