# VC-dimension of the infinite intersection of two spheres

I'm searching for an upper-bound for the VC-dimension of the infinite intersection of two spheres. Thanks

• How many spheres are interesting—two or infinity? And in what space? Finite-dimensional Euclidean or more exotic? Apr 20 at 14:21
• To be more precise, VC-dimension of the infinite phases of the moon. Each phase is the intersection of two spheres. In 2D
– shai
Apr 20 at 20:57

Once the OP has clarified that the question is about the VC-dimension of the 2-fold intersection of spheres in $$\mathbb{R}^d$$ (in fact, $$d=2$$ was specified), a simple upper bound can be stated. The VC-dim of spheres in $$\mathbb{R}^d$$ is the same as that of half-spaces (via an easy scaling argument) --- namely, $$d+1$$. Lemma 3.2.3 of Blumer et al. (1989) bounds the VC-dim of the $$k$$-fold union/intersection of a class of VC-dim $$d$$ by $$2kd\log(3k).$$ Thus, $$2\cdot2\cdot2\log(6)\approx 14$$ is an upper-bound on the VC-dimension in the OP.