Lemma: If P is family of d-dim, D-degree polynomials' nonneg sets, e.g. x^2-xy>=2, then VC-dim(P)<={d+D\choose d}.
Note: If D=1, then these are halfspaces and proof was homework.
Proof: Veronese map: Let m={d+D\choose d}-1, the number of monomials of degree at most D, except constant.
Associate each coordinate with one such monomial, e.g. if d=D=2, m=5, then (x,y,x^2,y^2,xy).
Define fi: R^2 -> R^5 in the natural way for points.
Also, for every D-degree poly, f, we have a naturally associated halfspace, h_f.
We have that f(p)>=0 if and only if fi(p) is in h_f, so we are done using D=1 special case.
Ex: Determine the VC-dim of discs in the plane.
For more complicated shapes, one can use Milnor-Thom (# of sign patterns<=(50Dn/d)^d for n polys).
Also, in general we have
Prop: Suppose R(X1,..,Xk) is a fixed relation using union, setminus etc. over k sets, like X1\X2 union X3.
If VC-dim(F)=d, then for G={R(X1,..,Xk):Xi is from F} we have VC-dim(G)=O(dklog k).
Proof: piG(m)<=(piF(m))^k and rest is calculation.
Hw: Determine the VC-dim of triangles in the plane.
Hw: Suppose VC-dim(F)=a and VC-dim(G)=b. At most how much can VC-dim(F union G) be? Upper and lower bounds!
Define dual set system: F*.
Lemma: VC-dim(F*)<2^{VC-dim(F)+1}.
Proof: We need that if VC-dim(F*)>=2^d, then VC-dim(F)>=d.
Take 2^d x 2^2^d matrix that shows a set of size 2^d shattered by F*.
In it we have a 2^d x d matrix that shows a set of size d shattered by F.
Art gallery problem - n/3 guards are always sufficient and sometimes necessary.
Thm: If there are no holes in the gallery (simply connected region) and from any point we can see 1/r of it (L-measure),
then at most Crlog r guards are enough.
Proof: Because of Epsilon Net Thm, it is enough to show that VC-dim of visibility regions is bounded.
Indirect suppose they shatter a point set of size d (where d is chosen large enough, to be specified later).
The d points determine <d^2 lines, these divide the plane into <d^4 regions.
Since there are 2^d guards that shatter, in a region we have >2^d/d^4 of these guards.
Since their shatter function is large on these d points, they also shatter a many points.
Using duality lemma, the points also shatter many guards.
Using same trick as before, we can suppose all points are in same region determined by lines of guards.
We needed this to ensure that guards can see the points in the same order and points can see guards in the same order.
If at beginning d was large enough, we have 5 guards, G1,..,G5, and 5 points, p1,..,p5, such that
pi can be seen by every Gj except for Gi, that cannot see it.
There is a pair, Gi-pi such that neither Gi, nor pi is the first or last in the order.
But then we have a hole in the gallery, a contradiction.
Ex: Neighborhood family of any planar graph has a bounded VC-dim.