Week 7
Ex0: Some real world application of Borsuk-Ulam that is better than stupid baricenteric map?
Ex1: Borsuk-Ulam is equivalent to its other version, where "antipodality is not required => there is a pair of antipodal points with same image".
Ex2: From Borsuk-Ulam prove that any continous function from a disc to itself has a fixed-point.
Week8
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.
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.