Conference rusure::math

704.0. "Ye Olde Geometry Problems" by CLT::GILBERT (eager like a child) Sat May 16 1987 23:25

1)  For a finite set S in the plane, let d(S) be the number of ways in which S
can be separated into two subsets by drawing a straight line.  For a fixed n,
what is the minimum and maximum of S for sets with n points?
 
2)  Let T be a tetrahedron in three dimensions.  Let A,B,C, and D be the areas
of its sides, and let a_AB,a_BC, and a_AC be the angles between the faces whose
areas are A,B, and C.  Find D as a function of A,B,C, and these three angles.
 
3)  Let T and T' be two regular simplices circumscribed about the unit sphere
in n dimensions which are oriented opposite to each other.
(i.e. if x is in T, -x is in T'). Estimate the volume of the intersection of T
and T' to within:
a) a factor of P(n), where P is a polynomial.
b) a constant factor.
 
4)  Show that if C is a convex body in the plane with unit area,
there is a triangle T containing C such that:
a)  T has area 4
b)  T has area 2
c)  T has area 2 and one side parallel to a given line l.
 
5)  Show that the regular dodecahedron exists and that there are six
Platonic solids in four dimensions.


[ The above problems were posted to sci.math by rutgers!ll-xn!husc6!endor!greg.
  The solutions have already been posted to sci.math, but I'll delay posting
  them here.
					- Gilbert ]