

Line 1: 
Line 1: 
−  The set function on a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013190/a0131901.png" /> equal to the area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013190/a0131902.png" /> of that part of the convex surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013190/a0131903.png" /> that has spherical image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013190/a0131904.png" />. This definition remains meaningful for general convex surfaces and it gives a totallyadditive set function on the ring of Borel sets.  +  {{TEXdone}} 
 +  The set function on a sphere $\Omega$ equal to the area $S(E)$ of that part of the convex surface $F$ that has spherical image $E\subset\Omega$. This definition remains meaningful for general convex surfaces and it gives a totallyadditive set function on the ring of Borel sets. 
   
 ====References====   ====References==== 
Line 7: 
Line 8: 
   
 ====Comments====   ====Comments==== 
−  In the article, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013190/a0131905.png" /> is the unit sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013190/a0131906.png" /> with centre at the origin. If one associates to each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013190/a0131907.png" /> the unit normal vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013190/a0131908.png" /> and shifts this vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013190/a0131909.png" /> so that its base coincides with the origin, then the end point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013190/a01319010.png" /> is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013190/a01319011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013190/a01319012.png" />. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013190/a01319013.png" /> is called the spherical image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013190/a01319014.png" />. The procedure for obtaining the spherical image of a point goes under the name of spherical mapping, cf. [[Spherical mapSpherical map]].  +  In the article, $\Omega$ is the unit sphere in $\mathbf R^3$ with centre at the origin. If one associates to each point $x\in F$ the unit normal vector $n_x$ and shifts this vector $n_x$ so that its base coincides with the origin, then the end point of $n_x$ is a point $x^*$ on $\Omega$. The point $x^*$ is called the spherical image of $x$. The procedure for obtaining the spherical image of a point goes under the name of spherical mapping, cf. [[Spherical mapSpherical map]]. 
Latest revision as of 15:31, 1 May 2014
The set function on a sphere $\Omega$ equal to the area $S(E)$ of that part of the convex surface $F$ that has spherical image $E\subset\Omega$. This definition remains meaningful for general convex surfaces and it gives a totallyadditive set function on the ring of Borel sets.
References
[1]  A.D. Aleksandrov, Mat. Sb. , 3 : 1 (1938) pp. 27–44 
[2]  H. Busemann, "Convex surfaces" , Interscience (1958) 
In the article, $\Omega$ is the unit sphere in $\mathbf R^3$ with centre at the origin. If one associates to each point $x\in F$ the unit normal vector $n_x$ and shifts this vector $n_x$ so that its base coincides with the origin, then the end point of $n_x$ is a point $x^*$ on $\Omega$. The point $x^*$ is called the spherical image of $x$. The procedure for obtaining the spherical image of a point goes under the name of spherical mapping, cf. Spherical map.
How to Cite This Entry:
Area function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Area_function&oldid=32073
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics  ISBN 1402006098.
See original article