# Tag Info

It has been recently shown by Csikos, Kupavskii, Mustafa in "Optimal Bounds on the VC-dimension" that the VC dimension of $k$-fold unions (or intersections or XORs) of half-spaces in $R^d$ behaves as $$\Theta(dk\log k).$$
Consider the function $f(x, y) = 1 - e^{-(x + y)}$. Now $f(0, 0) = 0$, $f$ is increasing and concave, since $g(t) = -e^{-t}$ is concave. But $f(1, 0) + f(0, 1) = 2(1 - e^{-1}) > 1 - e^{-2} = f(1, 1)$, hence the claim doesn't hold even when $n = 2$. The claim doesn't hold for convex functions either. $f(x, y) = 2(x + y) + |x - y|$ is one counterexample: \$...