Geodesic
The equivariant embeddings of $n$-dimensional space into the Hilbert space
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part VI, Tome 283 (2001), pp. 63-72 
M. Gorbulsky. The equivariant embeddings of $n$-dimensional space into the Hilbert space. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part VI, Tome 283 (2001), pp. 63-72. http://geodesic.mathdoc.fr/item/ZNSL_2001_283_a5/
@article{ZNSL_2001_283_a5,
     author = {M. Gorbulsky},
     title = {The equivariant embeddings of $n$-dimensional space into the {Hilbert} space},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {63--72},
     year = {2001},
     volume = {283},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_283_a5/}
}
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The present paper is devoted to the study of equivariant embeddings of $n$-dimensional space into the Hilbert space. We consider a representation of a group of similarities. The existence of a cocycle of this representation implies the existence of an isometric embedding of a metric group into the Hilbert space. Then we describe all cocycles of a representation of additive group of real numbers and construct an embedding of $n$-dimensional space supplied with a metric $d(x,y)=|x-y|^\alpha$ into the Hilbert space.