Roberts-type embeddings and conversion of transversal Tverberg's theorem

Abstract

Central in the paper are two results on the existence of `economical' embeddings in a Euclidean space. The first result (Corollary 1.4) states the existence of an embedding with image intersecting the large-dimensional planes in sets of `controllable' dimension. The second result (Corollary 1.6) proves the existence of maps such that each small-dimensional plane contains 'controllably' many points of the image. Well known results of Nöbeling-Pontryagin, Roberts, Hurewicz, Boltyanskii, and Goodsell can be obtained as consequences of these results. Their infinite-dimensional version concerning an embedding in a Hilbert space is also established (Theorem 1.8).