Consider a measurable convex bounded set \(K\subseteq \mathbb{R}^d\) with positive measure and \(0\in K\), and a function \(f\in L^p(K)\). Assume that \(f\) has an analytic extension on the polydisk
\[ \tilde{D}=D_1(0,6 \diam K)\times \cdots \times D_d(0,6\diam K). \]Then there is a constant \(C>0\) depending only on \(d\) such that for every measurable subset \(S\subseteq K\) with nonzero measure and \(p\ge 1\) the following estimate hold
\[ \lVert f\rVert_{L^p(K)}\le (C(\diam K)^d)^{2 \frac{\log M}{\log 2}} \lvert K\rvert^{1/p}\biggl(\frac{2}{\lvert S \rvert}\biggr)^{2\frac{\log M}{\log 2} + \frac{1}{p}} \lVert f\rVert_{L^p(S)} \]with
\[ M=\sup_{z\in \tilde{D}} \frac{\lvert K\rvert^{1/p}}{\lVert f\rVert_{L^p(K)}} \lvert f(z)\rvert. \]