\[ \DeclareMathOperator{\diam}{diam} \]

Suppose \(S\subseteq B\subseteq \mathbb{R}^d\) are bounded measurable sets with positive measure. For every point \(x\in B\) there is a line segment \(I\subset B\) starting in \(x\) with

\begin{equation*} \lvert S \rvert \le C (\diam B)^{d-1} \lvert S\cap I\rvert, \end{equation*}

where \(C>0\) is a constant depending only on \(d\).

Remarks
  • According to Kovrijkine [1] Kovrijkine this trick can be found in [2].
Proof

Using polar coordinates we may write

\begin{equation*} \lvert S\rvert = \int_{\mathbb{S}^{d-1}} \int_{0}^{\infty} 𝟙_{S}(x+r\xi)r^{d-1}\,dr\, d\xi. \end{equation*}

There is a direction \(\eta\in \mathbb{S}^{d-1}\) such that

\begin{equation*} \lvert S\rvert \le \sigma_{d-1} \int_{0}^{\infty} 𝟙_{S}(x+r\eta)r^{d-1}\,dr, \end{equation*}

where \(\sigma_{d-1}=\lvert \mathbb{S}^{d-1}\rvert\). Otherwise, one obtain a contradiction by integrating both sides over \(\mathbb{S}^{d-1}\). (Maxima cannot be smaller than the average.)

By estimating \(r\) on the right hand side with \(\diam B\) we obtain

\begin{equation*} \lvert S \rvert \le \sigma_{d-1} (\diam B)^{d-1} \lvert S\cap I\rvert. \end{equation*}

See also Link to heading

References Link to heading

  1. O. Kovrijkine, Some estimates of Fourier transforms, Ph.D. thesis, United States -- California: California Institute of Technology, 2000.
  2. F. Nazarov, Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type, Rossiĭskaya Akademiya Nauk. Algebra i Analiz, vol. 5, no. 4, p. 3–66, 1993.