Largest dual ellipsoids inscribed in dual cones

Suppose x? and s? lie in the interiors of a cone K and its dual K ^*, respectively. We seek dual ellipsoidal norms such that the product of the radii of the largest inscribed balls centered at x? and s? and inscribed in K and K ^*, respectively, is maximized. Here the balls are defined using the two dual norms. When the cones are symmetric, that is self-dual and homogeneous, the solution arises directly from the Nesterov–Todd primal–dual scaling. This shows a desirable geometric property of this scaling in symmetric cone programming, namely that it induces primal/dual norms that maximize the product of the distances to the boundaries of the cones.     

Clustering via minimum volume ellipsoids

We propose minimum volume ellipsoids (MVE) clustering as an alternative clustering technique to k-means for data clusters with ellipsoidal shapes and explore its value and practicality. MVE clustering allocates data points into clusters in a way that minimizes the geometric mean of the volumes of each cluster’s covering ellipsoids. Motivations for this approach include its scale-invariance, its ability to handle asymmetric and unequal clusters, and our ability to formulate it as a mixed-integer semidefinite programming problem that can be solved to global optimality. We present some preliminary empirical results that illustrate MVE clustering as an appropriate method for clustering data from mixtures of “ellipsoidal” distributions and compare its performance with the k-means clustering algorithm as well as the MCLUST algorithm (which is based on a maximum likelihood EM algorithm) available in the statistical package R. Research of the first author was supported in part by a Discovery Grant from NSERC and a research grant from Faculty of Mathematics, University of Waterloo. Research of the second author was supported in part by a Discovery Grant from NSERC and a PREA from Ontario, Canada.     

An inequality for M-matrices

If AB are n × n M matrices with dominant principal diagonal, we show that 3[det(A + B)]^1/n ≥ (det A)^1/n + (det B)^1/n.     

An isoperimetric inequality for the Heisenberg groups

We show that the Heisenberg groups of dimension five and higher, considered as Riemannian manifolds, satisfy a quadratic isoperimetric inequality. (This means that each loop of length L bounds a disk of area ~ L ².) This implies several important results about isoperimetric inequalities for discrete groups that act either on or on complex hyperbolic space, and provides interesting examples in geometric group theory. The proof consists of explicit construction of a disk spanning each loop in . Submitted: April 1997, Final version: November 1997     

An inequality for acute triangles

An inequality relating the tangents of half angles of a triangle that is necessary for solving Malfatti's problem is proved.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 11–14, 1992.     

An isoperimetric inequality for the Wiener sausage

Let (ξ(s)) _{s?≥ 0} be a standard Brownian motion in d?≥ 1 dimensions and let (D _s ) _{s ≥?0} be a collection of open sets in ${\mathbb{R}^d}$ . For each s, let B _s be a ball centered at 0 with vol(B _s ) =?vol(D _s ). We show that ${\mathbb{E}[\rm {vol}(\cup_{s \leq t}(\xi(s) + D_s))] \geq \mathbb{E}[\rm {vol}(\cup_{s \leq t}(\xi(s) + B_s))]}$ , for all t. In particular, this implies that the expected volume of the Wiener sausage increases when a drift is added to the Brownian motion.     

An inequality for Tutte polynomials

Let G be a graph without loops or bridges and a, b be positive real numbers with b ≥ a(a+2). We show that the Tutte polynomial of G satisfies the inequality T _G (b, 0)T _G (0, b) ≥ T _G (a, a)². Our result was inspired by a conjecture of Merino and Welsh that T _G (1, 1) ≤ max{T _G (2, 0),T _G (0, 2)}.     

An inequality for steiner systems

The following theorem is proved.Theorem. An inequalityv (t + 1)(k –t + 1) + (k –t) holds for Steiner systemsS(t, k, v) witht < k < v andt even with equality if and only if (t, k, v) = (t, t + 1, 2t + 3) or (4, 7, 23).     

An inequality for the modified Selberg zeta-function

The Ramanujan Journal - We consider the absolute values of the modified Selberg zeta-function at places symmetric with respect to the critical line. We prove an inequality for the modified Selberg...     

An inequality for the multivariate normal distribution

Herman Chernoff used Hermite polynomials to prove an inequality for the normal distribution. This inequality is useful in solving a variation of the classical isoperimetric problem which, in turn, is relevant to data compression in the theory of element identification. As the inequality is of interest in itself, we prove a multivariate generalization of it using a different argument.     

An inequality for generalized s-numbers

We shall prove an inequality for the generalized s-numbers of the product of two operators by a method which is new even in the case of matrices and compact operators.     

An inequality for 3-polytopes

We prove the inequality Σ_n≥7 (n−6) p_n≤v−12 for any 3-dimensional polytope with v vertices and p_n n-sided faces, such that Σ_n≥6 p_n≥3. The polytopes satisfying Σ_n≥7 (n−6) p_n=v−12 are described and an interpretation of our results is given in terms of density of n-sided faces in planar graphs.     

An inequality for non-negative matrices

Let A be an n × n matrix with non-negative entries and no entry in (0, 1). We prove that there exist integers r, s with 0 r s 2ⁿ such that A^r A^s. We prove that 2ⁿ cannot be replaced with e√n log n. We also give an application to the theory of formal languages.     

An inequality for self-inversive polynomials

A sharp L^p, p ? 1, inequality for the class of self-inversive polynomials is obtained.