Radii minimal projections of polytopes and constrained optimization of symmetric polynomials

We provide a characterization of the radii minimal projections of polytopes onto $j$-dimensional subspaces in Euclidean space \E^n. Applied on simplices this characterization allows to reduce the computation of an outer radius to a computation in the circumscribing case or to the computation of an outer radius of a lower-dimensional simplex. In the second part of the paper, we use this characterization to determine the sequence of outer $(n-1)$-radii of regular simplices (which are the radii of smallest enclosing cylinders). This settles a question which arose from the incidence that a paper by Wei{\ss}bach (1983) on this determination was erroneous. In the proof, we first reduce the problem to a constrained optimization problem of symmetric polynomials and then to an optimization problem in a fixed number of variables with additional integer constraints.Comment: Minor revisions. To appear in Advances in Geometr

Variance bounds, with an application to norm bounds for commutators

Murthy and Sethi (Sankhya Ser B \textbf{27}, 201--210 (1965)) gave a sharp upper bound on the variance of a real random variable in terms of the range of values of that variable. We generalise this bound to the complex case and, more importantly, to the matrix case. In doing so, we make contact with several geometrical and matrix analytical concepts, such as the numerical range, and introduce the new concept of radius of a matrix. We also give a new and simplified proof for a sharp upper bound on the Frobenius norm of commutators recently proven by B\"ottcher and Wenzel (Lin.\ Alg. Appl. \textbf{429} (2008) 1864--1885) and point out that at the heart of this proof lies exactly the matrix version of the variance we have introduced. As an immediate application of our variance bounds we obtain stronger versions of B\"ottcher and Wenzel's upper bound.Comment: 26 pages, 2 handmade drawing

Derivative relationships between volume and surface area of compact regions in R^d

We explore the idea that the derivative of the volume, V, of a region in R^d with respect to r equals its surface area, A, where r = d V/A. We show that the families of regions for which this formula for r is valid, which we call homogeneous families, include all the families of similar regions. We determine equivalent conditions for a family to be homogeneous, provide examples of homogeneous families made up of non-similar regions, and offer a geometric interpretation of r in a few cases.Comment: 15 page

Physical bounds and radiation modes for MIMO antennas

Modern antenna design for communication systems revolves around two extremes: devices, where only a small region is dedicated to antenna design, and base stations, where design space is not shared with other components. Both imply different restrictions on what performance is realizable. In this paper properties of both ends of the spectrum in terms of MIMO performance is investigated. For electrically small antennas the size restriction dominates the performance parameters. The regions dedicated to antenna design induce currents on the rest of the device. Here a method for studying fundamental bound on spectral efficiency of such configurations is presented. This bound is also studied for $N$-degree MIMO systems. For electrically large structures the number of degrees of freedom available per unit area is investigated for different shapes. Both of these are achieved by formulating a convex optimization problem for maximum spectral efficiency in the current density on the antenna. A computationally efficient solution for this problem is formulated and investigated in relation to constraining parameters, such as size and efficiency