Geodesic
Roberts-type embeddings and conversion of transversal Tverberg's theorem
Sbornik. Mathematics, Tome 196 (2005) no. 11, pp. 1585-1603

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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). 
@article{SM_2005_196_11_a1,
     author = {S. A. Bogatyi and V. M. Valov},
     title = {Roberts-type embeddings and conversion of transversal {Tverberg's} theorem},
     journal = {Sbornik. Mathematics},
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     publisher = {mathdoc},
     volume = {196},
     number = {11},
     year = {2005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2005_196_11_a1/}
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S. A. Bogatyi; V. M. Valov. Roberts-type embeddings and conversion of transversal Tverberg's theorem. Sbornik. Mathematics, Tome 196 (2005) no. 11, pp. 1585-1603. http://geodesic.mathdoc.fr/item/SM_2005_196_11_a1/