convex geometry In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbe ...

, the projection body

\Pi K

of a

convex body In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non-empty interior. A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in ...

K

in ''n''-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

is the convex body such that for any vector

u\in S^

, the

support function In mathematics, the support function ''h'A'' of a non-empty closed convex set ''A'' in \mathbb^n describes the (signed) distances of supporting hyperplanes of ''A'' from the origin. The support function is a convex function on \mathbb^n. Any n ...

\Pi K

in the direction ''u'' is the (''n'' – 1)-dimensional volume of the projection of ''K'' onto the

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

orthogonal to ''u''. Minkowski showed that the projection body of a convex body is convex. and used projection bodies in their solution to

Shephard's problem In mathematics, Shephard's problem, is the following geometrical question asked by Geoffrey Colin Shephard in 1964: if ''K'' and ''L'' are centrally symmetric convex bodies In mathematics, a convex body in n-dimensional Euclidean space \R^n is ...

. For

K

a convex body, let

\Pi^\circ K

denote the

polar body A polar body is a small haploid cell that is formed at the same time as an egg cell during oogenesis, but generally does not have the ability to be fertilized. It is named from its polar position in the egg. When certain diploid cells in animals ...

of its projection body. There are two remarkable affine isoperimetric inequality for this body. proved that for all convex bodies

K

, :

V_n(K)^ V_n(\Pi^\circ K)\le V_n(B^n)^ V_n(\Pi^\circ B^n),

where

B^n

denotes the ''n''-dimensional unit ball and

V_n

is ''n''-dimensional volume, and there is equality precisely for ellipsoids. proved that for all convex bodies

K

, :

V_n(K)^ V_n(\Pi^\circ K)\ge V_n(T^n)^ V_n(\Pi^\circ T^n),

where

T^n

denotes any

n

-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 In the mathematical field of integral geometry, the Funk transform (also known as Minkowski–Funk transform, Funk–Radon transform or spherical Radon transform) is an integral transform defined by integrating a function on great circles of the sp ...

of the radial function of ''K''. Intersection bodies were introduced by . 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 used this to show that the unit balls l, 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=Zhang , first1=Gaoyong , title=Restricted chord projection and affine inequalities , mr=1119653 , year=1991 , journal=

Geometriae Dedicata ''Geometriae Dedicata'' is a mathematical journal, founded in 1972, concentrating on geometry and its relationship to topology, group theory and the theory of dynamical systems. It was created on the initiative of Hans Freudenthal in Utrecht, the N ...

, volume=39 , issue=4 , pages=213–222, doi=10.1007/BF00182294 Convex geometry

See also

References