A power improvement in the Heilbronn triangle lower bound

There are absolute constants $\eta,c_1>0$ such that, for every sufficiently large integer $n$, one can choose $n$ points in the unit square so that every triangle they determine has area at least $c_1n^{-2+\eta}$. Thus the almost $n^{-2}$ upper-bound formulation of Heilbronn's triangle problem is false. The exponent $\eta$ is fixed but extremely small.