About the Isotropy Constant of Random Convex Sets
نویسنده
چکیده
K 〈x, θ〉2dx = LK ∀θ ∈ Sn−1 where LK is a constant independent of θ, which is called the isotropy constant of K. Here 〈·, ·〉 denotes the standard scalar product in R. It is well known that for every convex body K ∈ R there exists an affine map T such that TK is isotropic. Furthermore, K and TK are both isotropic if and only if T is an orthogonal transformation. In this case the isotropy constant of both K and TK is the same. Hence, we can define the isotropy constant for every convex body. It verifies the following equation:
منابع مشابه
Functionally 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 کاملSweep Line Algorithm for Convex Hull Revisited
Convex hull of some given points is the intersection of all convex sets containing them. It is used as primary structure in many other problems in computational geometry and other areas like image processing, model identification, geographical data systems, and triangular computation of a set of points and so on. Computing the convex hull of a set of point is one of the most fundamental and imp...
متن کاملAn embedding theorem for convex fuzzy sets
In this paper we embed the space of upper semicontinuous convex fuzzy sets on a Banach space into a space of continuous functions on a compact space. The following structures are preserved by the embedding: convex cone, metric, sup-semilattice. The indicator function of the unit ball is mapped to the constant function 1. Two applications are presented: strong laws of large numbers for fuzzy ran...
متن کاملConvex Generalized Semi-Infinite Programming Problems with Constraint Sets: Necessary Conditions
We consider generalized semi-infinite programming problems in which the index set of the inequality constraints depends on the decision vector and all emerging functions are assumed to be convex. Considering a lower level constraint qualification, we derive a formula for estimating the subdifferential of the value function. Finally, we establish the Fritz-John necessary optimality con...
متن کاملConvexity and Geodesic Metric Spaces
In this paper, we first present a preliminary study on metric segments and geodesics in metric spaces. Then we recall the concept of d-convexity of sets and functions in the sense of Menger and study some properties of d-convex sets and d-convex functions as well as extreme points and faces of d-convex sets in normed spaces. Finally we study the continuity of d-convex functions in geodesic metr...
متن کامل