Grant: University Faculty Award

This research focuses on the fifty year old Gohberg-Markus-Hadwiger Covering Conjecture, which was proposed in different guises, and independently by several mathematicians, between 1955 and 1963.  An object is called a convex body if the line joining any two points inside the object is completely contained within the object.  For example, a hockey stick is not a convex body, whereas a hockey puck is.  One formulation of the conjecture states that the minimum number of points (light sources) required to "illuminate" the entire boundary of a convex body in three dimensions is less than or equal to 8.  A point on the boundary of a convex body is called "illuminated" if the light beam passes from the outside of the body to the interior of the body at that point.  This means that a cube requires at least 8 light sources to illuminate it's boundary, one for each corner.   In general, the Gohberg-Markus-Hadwiger Covering Conjecture states that the minimum number of points required to illuminate the entire boundary of a convex body in d-dimensional space is less than or equal to 2 to the power d.  The conjecture is confirmed when the dimension, d, is equal to 2 and is, in general, open for d>2.  I believe the conjecture is true in general, as it has been verified for various classes of convex bodies in all dimensions.  I am currently trying to extend the conjecture to a wider class of convex bodies, called the neighbourly polytopes.  Researchers feel that extending the conjecture to these particular convex bodies would be an important milestone in the theory.