We consider the problem of finding a point in a nonempty bounded convex body Γ in the cone of symmetric positive semidefinite matrices S + . Assume that Γ is defined by a separating oracle, which, for any given m×m symmetric matrix Ŷ , either confirms that Ŷ ∈ Γ or returns several selected cuts, i.e., a number of symmetric matrices Ai, i = 1, ..., p, p ≤ pmax, such that Γ is in the polyhedron {...

A random polytope is the convex hull of uniformly distributed random points in a convex body K. A general lower bound on the variance of the volume and f -vector of random polytopes is proved. Also an upper bound in the case when K is a polytope is given. For polytopes, as for smooth convex bodies, the upper and lower bounds are of the same order of magnitude. The results imply a law of large n...

We characterise finite and infinitesimal rigidity for bar-joint frameworks in R with respect to polyhedral norms (i.e. norms with closed unit ball P a convex d-dimensional polytope). Infinitesimal and continuous rigidity are shown to be equivalent for finite frameworks in R which are well-positioned with respect to P. An edge-labelling determined by the facets of the unit ball and placement of ...

For a given family of matrices F , we present a methodology to construct (complex) polytope Barabanov norms (and, when F is nonnegative, polytope Barabanov monotone norms and antinorms). Invariant Barabanov norms have been introduced by Barabanov and constitute an important instrument to analyze the joint spectral radius of a set of matrices. In particular, they played a key role in the disprov...

The question of centers addresses the issue of how to inscribe an object within a region defined by a set of constraints. More than one centering approach can be defined which leads to a different inscribed object and a different derivation procedure for both the object as well as its center. When attempting to inscribe the largest sphere within the constraints polytope the problem is defined a...

The solution space of the rectangular linear system Az b, subject to x > 0, is called a polytope. An attempt is made to provide a deeper geometric insight, with numerical examples, into the condensed paper by Lord, et al. [1], that presents an algorithm to compute a center of a polytope. The algorithm is readily adopted for either sequential or parallel computer implementation. The computed cen...

a 50-year-old male underwent a check-up for an insurance company. chest x-ray revealed a large ball-like lesion in the posterior mediastinum. the finding was confirmed by computed tomography (ct) scan (figure 1). the patient was referred to our center, where he had a successful mass removal surgery (figure 2). pathology identified the mass as neurinoma.

For a given lattice, we establish an equivalence involving a closed zone of the corresponding Voronoi polytope, a lamina hyperplane of the corresponding Delaunay partition and a quadratic form of rank 1 being an extreme ray of the corresponding L-type domain. 1991 Mathematics Subject classification: primary 52C07; secondary 11H55 An n-dimensional lattice determines two normal partitions of the ...

We study the problem of covering a given set of n points in a high, d-dimensional space by the minimum enclosing polytope of a given arbitrary shape. We present algorithms that work for a large family of shapes, provided either only translations and no rotations are allowed, or only rotation about a fixed point is allowed; that is, one is allowed to only scale and translate a given shape, or sc...