| In the current monograph we discuss sets in the n-dimensional Euclidean space or in the Hilbert space l2 whose orthogonal projections onto all planes with codimension k, for a fixed natural number k, are convex. In particular, we consider sets with convex shadows, that is, the projections of sets onto hyperplanes. We address two main questions. The first questions is: What topological properties such sets must possess? The second question is whether we can find for every closed convex set B "minimal imitations" C of B, i.e. closed subsets C of B that have the same projections onto planes with codimension k for a fixed k, and are minimal with respect to dimension.; Regarding the above questions, we obtain the following main results. First, we obtain information about the structure of nonconvex compacta C in the n-dimensional Euclidean space (n ≥ 3) that have the property that every shadow is convex.; We find that if the dimension of C is less than n - 1 then there are n + 1 distinct hyperplanes whose intersections with C contain copies of (n - 2)-sphere.; If the dimension of C is greater than n - 2 then the existence of three and no more than three hyperplanes can be guaranteed.; Second, if C is a closed set with proper convex shadows then C contains a closed (n - 2)-dimensional manifold that is a product of a Euclidean space and a sphere.; Third, for every n-dimensional closed convex set B we construct a "minimal imitation" C of B such that C and B have the same projections onto all k-planes, for a fixed natural k.; Finally, we extend some of our results in the Hilbert space l2. We show that if k is a natural number, B ⊂ l2 is closed and convex with nonempty interior, C is a closed set and B and C have identical proper projections onto every plane with codimension k, then C must contain a closed copy of Hilbert space. We also show that if a compact set has convex projections onto all first n factors then it must be convex. |
