  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\'ar\'any-Katchalski-Pach and
Nasz\'odi. 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.
