Directed projection functions of convex bodies

For 1 ≤ i < j < d, a j-dimensional subspace L of ℝd and a convex body K in ℝd, we consider the projection K|L of K onto L. The directed projection function v i,j (K;L,u) is defined to be the i-dimensional size of the part of K|L which is illuminated in direction u L. This involves the i-th surface area measure of K|L and is motivated by Groemer's [17] notion of semi-girth of bodies in ℝd. It is well-known that centrally symmetric bodies are determined (up to translation) by their projection functions, we extend this by showing that an arbitrary body is determined by any one of its directed projection functions. We also obtain a corresponding stability result. Groemer [17] addressed the case i = 1, j = 2, d = 3. For j > 1, we then consider the average of v 1,j (K;L,u) over all spaces L containing u and investigate whether the resulting function ν̄1,j(K,u) determines K. We will find pairs (d,j) for which this is the case and some pairs for which it is false. The latter situation will be seen to be related to some classical results from number theory. We will also consider more general averages for the case of centrally symmetric bodies.