Shadow-boundaries of convex bodies Original Research Article

We shall summarize results from geometric convexity referring to sharp shadow-boundaries of convex polytopes (convex bodies) in Rd, d ⩾ 2. Such shadow-boundaries are homeomorphic to (r − 1)-spheres and can be represented as intersections of these polytopes (bodies) with (d - r)-dimensional supporting flats forming supporting cylinders or cones, where 1 ⩽ r ⩽ d - 1. The results discussed mainly belong to the combinatorial theory of convex polytopes, and it is shown how these statements can be applied to solve problems of a different nature (such as covering problems, the description of projection functions, etc.)