linear subspaces def: closed under 1) scaling 2) linear combination.
Linear span of a set of points: intersection of all linear spaces containing it
Alternate algebraic formulation: set of all finite linear combinations.
Linear dependence
Note: always contains the origin.
Affine subspaces : translated linear spaces.
Affine span: intersection of all affine spaces containing the set
Alternate algebraic formulation: set of all 'affine' combinations.
Affine dependence
Convex sets: line segment between any two points is contained in the set
Convex hull: intersection of all convex sets containing the points
algebraic formulation: set of all 'convex' combinations
Caratheodory's theorem. sketch of incorrect proof. sketch of correct proof.
Radon's lemma.
Use of Radon's lemma: Helly's theorem.
HW1: Given a finite set of points X in the plane with diameter 1 (the distance between any two points is atmost 1), there exists a circle of radius 1/sqrt(3) containing all the points in X.
a) Prove the statement when there are exactly 3 points in X
b) Prove it for a general finite set X