Convolutions and Multiplier Transformations of Convex Bodies
نویسنده
چکیده
Rotation intertwining maps from the set of convex bodies in R into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An application of this representation is a complete classification of all even BlaschkeMinkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator. Among further applications is the following result: If an even Blaschke-Minkowski homomorphism maps a convex body to a polytope, then it is a constant multiple of the projection body operator.
منابع مشابه
Some properties of extended multiplier transformations to the classes of meromorphic multivalent functions
In this paper, we introduce new classes $sum_{k,p,n}(alpha ,m,lambda ,l,rho )$ and $mathcal{T}_{k,p,n}(alpha ,m,lambda ,l,rho )$ of p-valent meromorphic functions defined by using the extended multiplier transformation operator. We use a strong convolution technique and derive inclusion results. A radius problem and some other interesting properties of these classes are discussed.
متن کاملFunctionally closed sets and functionally convex sets in real Banach spaces
Let $X$ be a real normed space, then $C(subseteq X)$ is functionally convex (briefly, $F$-convex), if $T(C)subseteq Bbb R $ is convex for all bounded linear transformations $Tin B(X,R)$; and $K(subseteq X)$ is functionally closed (briefly, $F$-closed), if $T(K)subseteq Bbb R $ is closed for all bounded linear transformations $Tin B(X,R)$. We improve the Krein-Milman theorem ...
متن کاملOn the Algebra of Intervals and Convex Bodies
We introduce and study abstract algebraic systems generalizing the arithmetic systems of intervals and convex bodies involving Minkowski operations such as quasimodules and quasilinear systems. Embedding theorems are proved and computational rules for algebraic transformations are given.
متن کاملNo return to convexity
In the paper we study closures of classes of log–concave measures under taking weak limits, linear transformations and tensor products. We consider what uniform measures on convex bodies can one obtain starting from some class K. In particular we prove that if one starts from one–dimensional log–concave measures, one obtains no non– trivial uniform mesures on convex bodies.
متن کاملA Characterization of the Entropy--Gibbs Transformations
Let h be a finite dimensional complex Hilbert space, b(h)+ be the set of all positive semi-definite operators on h and Phi is a (not necessarily linear) unital map of B(H) + preserving the Entropy-Gibbs transformation. Then there exists either a unitary or an anti-unitary operator U on H such that Phi(A) = UAU* for any B(H) +. Thermodynamics, a branch of physics that is concerned with the study...
متن کامل