22 I. Convex Sets at Large is non-empty. Since the sets Ar and Ab are open, their intersection is an open set therefore an intersection of sets Ar and Ab is non-empty if and only if it contains a point (c, α) with c = 0. Hence for any subset S R B, the intersection r∈S∩R Ar b∈S∩B Ab is not empty if and only if there is a point (c, α) Rd+1 such that the hyperplane H = x Rd : c, x = α strictly separates the sets B S and R S. This completes the proof. For example, if two sets of points in the plain cannot be separated by a straight line, one of the three configurations of Figure 6 must occur. 1 ) 2 ) 3 ) Figure 6. The three reasons points cannot be separated in the plane PROBLEMS. 1. Prove that if a convex set is contained in the union of a finite family of halfspaces in Rd (sometimes we say covered by a finite family of halfspaces see Section 5.2), then it is contained in the union of some d + 1 (or fewer) halfspaces from the family (covered by some d + 1 subspaces). 2. Let I1,... , Im be parallel line segments in R2, such that for every three Ii 1 , Ii 2 , Ii 3 there is a straight line that intersects all three. Prove that there is a straight line that intersects all the segments I1,... , Im. 3. Let Ai : i = 1, . . . , m be convex sets in R2 such that for every two sets Ai and Aj there is a line parallel to the x-axis which intersects them both. Prove that there is a line parallel to the x-axis which intersects all the sets Ai. (5.2) The center point. Let us fix a Borel probability measure μ on Rd. This means, roughly speaking, that for any “reasonable” subset A Rd a non-negative number μ(A) is assigned which satisfies some additivity and continuity properties and such that μ(Rd) = 1. We are not interested in rigorous definitions here the following two examples are already of interest:
Previous Page Next Page