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Difference between revisions of "Area function"

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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 totally-additive set function on the ring of Borel sets.
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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 totally-additive set function on the ring of Borel sets.
  
 
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====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 map|Spherical map]].
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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|Spherical 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 totally-additive 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)


Comments

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=11860
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article