Let K be a smooth convex set with volume one in R. Choose n random points in K independently according to the uniform distribution. The convex hull of these points, denoted by Kn, is called a random polytope. We prove that several key functionals of Kn satisfy the central limit theorem as n tends to infinity.

In this note, we prove the following result. Let K ⊂ Rd be a convex body with the origin O in its interior. If there is a number λ ∈ (0, 1) such that the n-dimensional volume of the convex hull of the union of K with the translates of λK , by a vector x , depends only on the Euclidean norm of x , then K is a Euclidean ball.

Radon’s theorem asserts that any set S of d + 2 points in [Wd has a partition into two subsets S, and S, such that Conv(S,) rl Conv(S,) # 0, where Conv(Si) denotes the convex hull of Si. This central theorem in the theory of convexity has been extended in many directions. One of the most interesting generalizations is the following theorem, proved by Tverberg in 1966, and now considered as a cl...

The evenly convex hull of a given set is the intersection of all the open halfspaces which contain such set (hence the convex hull is contained in the evenly convex hull). This paper deals with nite dimensional linear systems containing strict inequalities and (possibly) weak inequalities as well as equalities. The number of inequalities and equalities in these systems is arbitrary (possibly in...

The mathematical principles come from the "Maximum Likelihood Method" [2, 3], used in probability theory for the determination of distribution parameters from experimental data. The Maximum Likelihood analysis leads to the definition of the Probabilistic Hough Transform, which is a likelihood function. If certain assumptions are made about the error characteristics, the PHT is very close to con...