This is an announcement for the paper "Brascamp-Lieb inequality and quantitative versions of Helly's theorem" by Silouanos Brazitikos. Abstract: We provide a number of new quantitative versions of Helly's theorem. For example, we show that for every family $\{P_i:i\in I\}$ of closed half-spaces $$P_i=\{ x\in {\mathbb R}^n:\langle x,w_i\rangle \leq 1\}$$ in ${\mathbb R}^n$ such that $P=\bigcap_{i\in I}P_i$ has positive volume, there exist $s\leq \alpha n$ and $i_1,\ldots , i_s\in I$ such that $$|P_{i_1}\cap\cdots\cap P_{i_s}|\leq (Cn)^n\,|P|,$$ where $\alpha , C>0$ are absolute constants. These results complement and improve previous work of B\'{a}r\'{a}ny-Katchalski-Pach and Nasz\'{o}di. Our method combines the work of Srivastava on approximate John's decompositions with few vectors, a new estimate on the corresponding constant in the Brascamp-Lieb inequality and an appropriate variant of Ball's proof of the reverse isoperimetric inequality. Archive classification: math.FA Mathematics Subject Classification: 26D15 Submitted from: silouanb@math.uoa.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.05783 or http://arXiv.org/abs/1509.05783