View source for Ray function

A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777001.png" /> defined on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777002.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777003.png" /> and satisfying the following conditions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777004.png" /> is continuous, non-negative and homogeneous (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777005.png" /> for any real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777006.png" />). A ray function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777007.png" /> is said to be positive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777008.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777009.png" />, and symmetric if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770010.png" />. A ray function is said to be convex if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770011.png" />,

<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/r/r077/r077700/r07770012.png" /></td> </tr></table>

For any ray function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770013.png" /> there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770014.png" /> for which

<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/r/r077/r077700/r07770015.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770016.png" /> is positive, then there is also a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770017.png" /> for which

<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/r/r077/r077700/r07770018.png" /></td> </tr></table>

The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770019.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770020.png" /> satisfying the condition

<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/r/r077/r077700/r07770021.png" /></td> </tr></table>

is a [[Star body|star body]]. Conversely, for any open star body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770022.png" /> there is a unique ray function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770023.png" /> for which

<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/r/r077/r077700/r07770024.png" /></td> </tr></table>

A star body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770025.png" /> is bounded if and only if its ray function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770026.png" /> is positive. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770027.png" /> is a symmetric function, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770028.png" /> is symmetric about the point 0; the converse is also true. A star body is convex if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770029.png" /> is a convex ray function.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.W.S. Cassels,   "An introduction to the geometry of numbers" , Springer  (1959)</TD></TR></table>



====Comments====
Star bodies are usually defined as closed ray sets. A ray function is more commonly called a distance function.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Gruber,   C.G. Lekkerkerker,   "Geometry of numbers" , North-Holland  (1987)  pp. Sect. (iv)  (Updated reprint)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Hlawka,   "Das inhomogene Problem in der Geometrie der Zahlen" , ''Proc. Internat. Congress Mathematicians (Amsterdam, 1954)'' , '''3''' , Noordhoff  (1954)  pp. 20–27  ((Also: Selecta, Springer 1990, 178–185.))</TD></TR></table>