Let $F_1,\ldots , F_s\in \mathbb {Z}[X_1,\ldots ,X_n]$ be quasiconvex polynomials of degrees bounded by $d\geq 2$ and assume that the maximum binary length of the coefficients of these polynomials doesn't exceed a given natural number $\ell $. We show that the system of polynomial inequalities $F_1\leq 0,\ldots ,F_s\leq 0$ admits an integer solution if and only if such a solution with binary length bounded by $(sd)^{cn}\ell $ exists. (Here $c$ is a constant, independent on $s,d,n$ and $\ell $). The simply exponential nature of this bound is intrinsical to this problem. We obtain a similar geometrical bound for the corresponding minimisation problem.