“Projection body”的意思、由来-开放百科全书

词条

Projection body

释义

See also
References

In convex geometry, the projection body of a convex body in n-dimensional Euclidean space is the convex body such that for any vector , the support function of in the direction u is the (n – 1)-dimensional volume of the projection of K onto the hyperplane orthogonal to u.

Minkowski showed that the projection body of a convex body is convex. {{harvtxt|Petty|1967}} and {{harvtxt|Schneider|1967}} used projection bodies in their solution to Shephard's problem.

For a convex body, let denote the polar body of its projection body. There are two remarkable affine isoperimetric inequality for this body. {{harvtxt|Petty|1971}} proved that for all convex bodies ,

where denotes the n-dimensional unit ball and is n-dimensional volume, and there is equality precisely for ellipsoids. {{harvtxt|Zhang|1991}} proved that for all convex bodies ,

where denotes any -dimensional simplex, and there is equality precisely for such simplices.

The intersection body IK of K is defined similarly, as the star body such that for any vector u the radial function of IK from the origin in direction u is the (n – 1)-dimensional volume of the intersection of K with the hyperplane u^⊥.

Equivalently, the radial function of the intersection body IK is the Funk transform of the radial function of K.

Intersection bodies were introduced by {{harvtxt|Lutwak|1988}}.

{{harvtxt|Koldobsky|1998a}} showed that a centrally symmetric star-shaped body is an intersection body if and only if the function 1/||x|| is a positive definite distribution, where ||x|| is the homogeneous function of degree 1 that is 1 on the boundary of the body, and {{harvtxt|Koldobsky|1998b}} used this to show that the unit balls l{{su|b=n|p = p}}, 2 < p ≤ ∞ in n-dimensional space with the l^p norm are intersection bodies for n=4 but are not intersection bodies for n ≥ 5.

References

{{Citation | last1=Bourgain | first1=Jean | last2=Lindenstrauss | first2=J. | title=Geometric aspects of functional analysis (1986/87) | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Math. | doi=10.1007/BFb0081746 |mr=950986 | year=1988 | volume=1317 | chapter=Projection bodies | pages=250–270| isbn=978-3-540-19353-1 }}
{{Citation | last1=Koldobsky | first1=Alexander | title=Intersection bodies, positive definite distributions, and the Busemann-Petty problem |mr=1637955 | year=1998a | journal=American Journal of Mathematics | issn=0002-9327 | volume=120 | issue=4 | pages=827–840 | doi=10.1353/ajm.1998.0030| citeseerx=10.1.1.610.5349 }}
{{Citation | last1=Koldobsky | first1=Alexander | title=Intersection bodies in R⁴ | doi=10.1006/aima.1998.1718 |mr=1623669 | year=1998b | journal=Advances in Mathematics | issn=0001-8708 | volume=136 | issue=1 | pages=1–14}}
{{Citation | last1=Lutwak | first1=Erwin|authorlink1=Erwin Lutwak | title=Intersection bodies and dual mixed volumes | doi=10.1016/0001-8708(88)90077-1 |mr=963487 | year=1988 | journal=Advances in Mathematics | issn=0001-8708 | volume=71 | issue=2 | pages=232–261}}
{{Citation | last1=Petty | first1=Clinton M. | title=Proceedings of the Colloquium on Convexity (Copenhagen, 1965) | chapter-url=https://books.google.com/books?id=UIjyTgEACAAJ | publisher=Kobenhavns Univ. Mat. Inst., Copenhagen |mr=0216369 | year=1967 | chapter=Projection bodies | pages=234–241}}
{{Citation | last1=Petty | first1=Clinton M. | title=Proceedings of the Conference on Convexity and Combinatorial Geometry (Univ. Oklahoma, Norman, Okla., 1971). Dept. Math., Univ. Oklahoma, Norman, Oklahoma | mr=0362057 | year=1971 | chapter=Isoperimetric problems| pages=26–41}}
{{cite journal| last = Schneider| first = Rolf| title = Zur einem Problem von Shephard über die Projektionen konvexer Körper| journal = Math. Z.| volume = 101| year = 1967| pages = 71–82| language = German| doi = 10.1007/BF01135693}}
{{Citation | last1=Zhang | first1=Gaoyong | title=Restricted chord projection and affine inequalities |mr=1119653 | year=1991 | journal=Geometriae Dedicata | volume=39 | issue=4 | pages=213–222| doi=10.1007/BF00182294 }}

1 : Convex geometry

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。

See also

References