Isomorphic embeddings of finite
metric spaces into Hamming spaces

B. Olijnyk

The category of finite metric spaces, morphisms of which are the~maps preserving unequalities of distances between pairs of points, is considered. It is proved that every finite metric space admits an isomorphic embedding (in sense of this category) into some Hamming space. The obvious construction of such an embedding is indicated.

1. \parskip1pt \Refsfrills{. Deza M., Shpectorov S. Recognition of the $l_1$-graphs with complexity $O(nm)$, or football in a hypercube // Rapport de Recherche du LIENS-95-11 --1995 p.1--14

2. Frucht R. Herstellung von Graphen mit vorgegebener abstracten Gruppe// Compositio Math.–1938 V.6 p.239–250

3. Olijnyk B. Isomorphic embeddings of finite metric spaces into Hamming spaces// Theses of reports of conference “Mathematical methods and its applications”, Lviv– 1995 p.39–40 Ukrainian

4. Ganyushkin A.G., Tsvirkunov V.V. On classification of finite metric spaces// (Translated from Matematicheskie Zametki(in Russian), Mathematical Notes– 1994 V. 56 p.1023–1029

5. Schoenberg I.J. Metric spaces and completely monotone functions// Ann. Math.– 1938 V. 39 p.811–841

6. Blumental L.M. Theory and applications of distance geometry ,addr New York, Chelsea– 1970

7. Maehara H. Metric transforms of finite spaces and connected graphs // Disk. Math.– 1986 V. 61 p.235–246

8. Fichet B Dimensionality problems in $L_1$-norm representations \inbook Van Cutsem, B. (Ed.) Classification and dissimilarity analysis ,addr New York, Springer-Verlag, Lecture Notes in Statistics-- 1994 p.201--224

	Isomorphic embeddings of finite metric spaces into Hamming spaces
Author	B. Olijnyk
Abstract	The category of finite metric spaces, morphisms of which are the~maps preserving unequalities of distances between pairs of points, is considered. It is proved that every finite metric space admits an isomorphic embedding (in sense of this category) into some Hamming space. The obvious construction of such an embedding is indicated.
Keywords	finite metric spaces, morphisms, distance inequalities, isomorphic embedding, Hamming space
DOI	doi:10.30970/ms.8.2.176-179
Reference	1. \parskip1pt \Refsfrills{. Deza M., Shpectorov S. Recognition of the $l_1$-graphs with complexity $O(nm)$, or football in a hypercube // Rapport de Recherche du LIENS-95-11 --1995 p.1--14 2. Frucht R. Herstellung von Graphen mit vorgegebener abstracten Gruppe// Compositio Math.–1938 V.6 p.239–250 3. Olijnyk B. Isomorphic embeddings of finite metric spaces into Hamming spaces// Theses of reports of conference “Mathematical methods and its applications”, Lviv– 1995 p.39–40 Ukrainian 4. Ganyushkin A.G., Tsvirkunov V.V. On classification of finite metric spaces// (Translated from Matematicheskie Zametki(in Russian), Mathematical Notes– 1994 V. 56 p.1023–1029 5. Schoenberg I.J. Metric spaces and completely monotone functions// Ann. Math.– 1938 V. 39 p.811–841 6. Blumental L.M. Theory and applications of distance geometry ,addr New York, Chelsea– 1970 7. Maehara H. Metric transforms of finite spaces and connected graphs // Disk. Math.– 1986 V. 61 p.235–246 8. Fichet B Dimensionality problems in $L_1$-norm representations \inbook Van Cutsem, B. (Ed.) Classification and dissimilarity analysis ,addr New York, Springer-Verlag, Lecture Notes in Statistics-- 1994 p.201--224
Pages	176-179
Volume	8
Issue	2
Year	1997
Journal	Matematychni Studii
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