Convexity radius
convexity limit, , of a function
The least upper bound of the radii of the spheres
, each one of which is mapped into a convex domain; here, the function
is defined on a domain
of a metric space with metric
and assumes values in a linear space. The convexity radius
at a point
with respect to some class
of mappings
of the domain
is, by definition, the number
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If is an affine mapping of a Euclidean space
,
, then
. With respect to the class
of all normalized univalent conformal mappings
,
,
, of the unit disc
of the complex plane, the convexity radius equals
; if one imposes the additional condition of convexity of the domains
, i.e. for convex functions (cf. Convex function (of a complex variable)),
. See also Univalent function.
References
[1] | G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) |
[2] | I.A. Aleksandrov, "On bounds for convexity and starlikeness of functions univalent and regular in a disc" Dokl. Akad. Nauk SSSR , 116 : 6 (1957) pp. 903–905 (In Russian) |
[3] | A. Marx, "Untersuchungen über schlichte Abbildungen" Math. Ann. , 107 (1932) pp. 40–67 |
Convexity radius. I.A. Aleksandrov (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convexity_radius&oldid=14892