View source for Convexity radius

''convexity limit, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c0264201.png" />, of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c0264202.png" />''

The least upper bound of the radii <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c0264203.png" /> of the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c0264204.png" />, each one of which is mapped into a [[Convex domain|convex domain]]; here, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c0264205.png" /> is defined on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c0264206.png" /> of a metric space with metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c0264207.png" /> and assumes values in a linear space. The convexity radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c0264208.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c0264209.png" /> with respect to some class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c02642010.png" /> of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c02642011.png" /> of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c02642012.png" /> is, by definition, the number

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c02642013.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c02642014.png" /> is an affine mapping of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c02642015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c02642016.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c02642017.png" />. With respect to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c02642018.png" /> of all normalized univalent conformal mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c02642019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c02642020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c02642021.png" />, of the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c02642022.png" /> of the complex plane, the convexity radius equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c02642023.png" />; if one imposes the additional condition of convexity of the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c02642024.png" />, i.e. for convex functions (cf. [[Convex function (of a complex variable)|Convex function (of a complex variable)]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026420/c02642025.png" />. See also [[Univalent function|Univalent function]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Marx,   "Untersuchungen über schlichte Abbildungen"  ''Math. Ann.'' , '''107'''  (1932)  pp. 40–67</TD></TR></table>