Semiconvex compacta
Nykyforchyn, Oleh R.
Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997), p. 761-774 / Harvested from Czech Digital Mathematics Library
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We define and investigate a generalization of the notion of convex compacta. Namely, for semiconvex combination in a semiconvex compactum we allow the existence of non-trivial loops connecting a point with itself. It is proved that any semiconvex compactum contains two non-empty convex compacta, the center and the weak center. The center is the largest compactum such that semiconvex combination induces a convex structure on it. The convex structure on the weak center does not necessarily coincide with the structure induced by semiconvex combination but generates the latter in a special manner. A sufficient condition for a net of semiconvex combinations to converge to the weak center (``the law of large numbers'') is established. A semiconvex compactum is called strongly semiconvex if its center and its weak center coincide. Some natural constructions of topology and functional analysis are shown to be (strongly) semiconvex compacta. It is shown that the construction of center is functorial and gives the reflector that is the left adjoint to the embedding of the category of convex compacta into the category of strongly semiconvex compacta. Also the left adjoint to the forgetful functor from the category of strongly semiconvex compacta to the category of compacta is constructed.

Publié le : 1997-01-01
Classification:  18B30,  18B40,  52A01,  54B30,  54D30 
@article{118971,
     author = {Oleh R. Nykyforchyn},
     title = {Semiconvex compacta},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {38},
     year = {1997},
     pages = {761-774},
     zbl = {0938.54011},
     mrnumber = {1603714},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118971}
}

Nykyforchyn, Oleh R. Semiconvex compacta. Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) pp. 761-774. http://gdmltest.u-ga.fr/item/118971/

Adams J.F. Lectures on Lie Groups, W.A. Benjamin Inc, New York-Amsterdam, 1969. | MR 0252560 | Zbl 0515.22007

Eilenberg S.; Moore J.C. Adjoint functors and triples, Ill. Univ. J. Math. 9 (1965), 381-398. (1965) | MR 0184984 | Zbl 0135.02103

Engelking R. General Topology, PWN, Warszawa, 1977. | MR 0500780 | Zbl 0684.54001

Fedorchuk V.V. Probability measures in topology (in Russian), Uspekhi Mat. Nauk 46 (1991), 41-80. (1991) | MR 1109036

Fedorchuk V.V.; Filippov V.V. General Topology: Basic Constructions (in Russian), Izd-vo MGU, Moscow, 1988.

Melnikov O.V.; Remeslennikov V.N.; Romankov V.A. Et Al. General Algebra (in Russian), vol. 2, Skornyakov L.A. (ed), Nauka, 1990. | MR 1137273

Glushkov V.M. Structure of locally bicompact groups and the fifth problem of Hilbert (in Russian), Uspekhi Mat. Nauk 12 (1957), 3-41. (1957) | MR 0101892

Kantorovich L.V.; Rubinstein G.Sh. On one functional space in some extremal problems (in Russian), Dokl. Akad. Nauk. SSSR 115 (1957), 1058-1061. (1957) | MR 0094707

Świrszcz T. Monadic functors and convexity, Bull. Acad. Pol. Sci., Sér. Sci. Math., Astr. et Phys. 22 (1974), 39-42. (1974) | MR 0390019