Polynomial Cutting lemma: If we have m points in the plane, we can partition them with the zero-curve (P=0)
of a degree D polynomial such that each of the resulting O(D^2) cells has at most O(m/D^2) points.
From here we can again prove Szemeredi-Ttrotter using its weak version, O(mn^1/2+n).
Proof: Each line can intersect P=0 at most D times (unless it is contained in it).
Number of incidences in each cell is less than O(m_i n_i^1/2 +n_i).
We have for each i (there are D^2 of them) that m_i <= m/D^2 and sum_i n_i <= nD.
From here incidences in all cells <= D^(-1/2)mn^1/2 + Dn.
To optimize we choose D=m^2/3 n^(-1/3).
We also have to count incidences on P=0.
Every line is either contained in P=0, or it intersects it at most D times.
Latter type give only Dn incidences, so we have to bound the former.
There are at most D lines contained in P=0: Linear factors can be taken out from two valued polynomials.
If D<n/2, then we can use induction.
If D>n/2, then n^2=O(m), so O(m) is the main part, which follows from the (dual) weak Sz-T.
Convex polytopes.
two definitions: V-polytope is a convex hull of finitely many points. H-polyhedron is the intersection of finitely many half spaces. If bounded, a H-polyhedron is called an H-polytope.
Thm: every H-polytope is a V-polytope and vice versa.
examples of polytopes: a cube, cross polytope, simplex, permutahederon.
The n-1 dimensional permutahederon can be best described in R^n as the convex hull of points obtained by every permutation of the coordinates of the point (1,2,3,...,n).
Duality of point and hyperplane. dual of a point 'a' is the hyperplane <a,x> = 1 denoted by D_0(a).
another way to understand this duality is to lift it to d+1 dimensions. Dual of a k-flat (an affine k dimensional subspace) is the intersection of its orthogonal complement with the x_(d+1) = -1 plane.
Duality preserves incidences:
a) if p \in h, then D_0(h) \in D_0(p).
b) if p \in h- then D_0(h) \in D_0(p)-.
where h- denotes the half space bounded by hyperplane h which contains the origin.
what is the dual of a point on the line segment joining two points?
Polar of a set X, denoted X* = {y: <x,y> <= 1 for all x \in X}.
eg: polar of a point?
eg2: polar of a wedge containing the origin
eg3: polar of a closed convex set containing the origin
eg4: polar of a triangle not containing the origin
Ex1: show that for any conv set (X*)* = closure(conv(x \cup 0)).
Ex2: show that C* is bounded iff C contains the origin.
Ex3: show that C=C* iff its a ball.
Ex4: a) what is the dual of all lines that intersect a segment?Â
b) what if the line through the segment passes through the origin?
c) given a set of n segment s.t. the line containing each passes through the origin and that for every three segments, there is a line that stabs them, show that there is a line that stabs all.
d) for every n, find a collection of n+1 segments such that for every n subset, there is a line that stabs them, but there is no line that stabs all.
Face of a polytope P is the intersection of a hyperplane h with P where P is contained in either h- or h+.
Properties of faces of a polytope:
a) Face of a face is a face.
b) Vertices are extremal.
Nomenclature:
0-faces: vertices, 1-faces: edges, d-1-faces: facets.
The graph obtained from the vertices and edges is called the graph of the polytope.
Steinitz thm: a finite graph is isomorphic to the graph of a 3 dimensional polytope iff it is a planar 3-connected graph.
HW: Permutahederon.
a) show that it is n-1 dimensional (we discussed why it is at most n-1 dimensional in class, here you have to show why exactly n-1 dimensional).
b) Show that it has n! vertices
c) describe all its faces
d) determine the dimensions of the faces found. In particular, show that facets correspond to the ordered partitions (A,B) of [n], A,B non empty.
Ex5: a) what is the polar of the standard cube (vertices are [+-1]^n)?
b) the polar of the standard crosspolytope (vertices are +-e_i)?
c) If P is a 3 dimensional polytope and G(P) denotes its graph, is there a relationship between G(P) and G(P*)?
Ex6: Show that every symmetric polytope is an affine image of a cross polytope of higher dimension.