Power savings for planar halving lines and k-sets

Improves the planar halving-line bound to $O(n^{4/3-\varepsilon})$ for sets with no three collinear and an absolute ε > 0. More generally, an n-point set with no three collinear has $O(n(k+1)^{1/3-\varepsilon_0})$ strictly separable k-subsets for $1\le k\le n/2$, with an absolute $\varepsilon_0\gt 0$. The constants and positive exponents are nonquantitative.