Introduction 2 1. Normed Rings 4 1.1. Elementwise Norms 4 1.2. Ideal norms 5 1.3. Finite Quotient Domains 5 1.4. Euclidean norms 6 2. Euclidean quadratic forms and ADC forms 7 2.1. Euclidean quadratic forms 7 2.2. Euclideanity 7 2.3. ADC-forms 8 2.4. Euclidean Forms are ADC Forms 8 2.5. The Generalized Cassels-Pfister Theorem 9 2.6. Maximal Lattices 10 3. Localization and Completion 11 3.1. Loc...

This analysis introduces a generalization of the basic statistical concepts of expectation values and variance for non-Euclidean metrics induced by L-norms. The non-Euclidean L means are defined by exploiting the fundamental property of minimizing the L deviations that compose the L variance. These L expectation values embody a generic formal scheme of means characterization. Having the p-norm ...

This analysis introduces a generalization of the basic statistical concepts of expectation values and variance for non-Euclidean metrics induced by L-norms. The non-Euclidean L means are defined by exploiting the fundamental property of minimizing the L deviations that compose the L variance. These L expectation values embody a generic formal scheme of means characterization. Having the p-norm ...

An algorithm for minimizing a sum of Euclidean Norms subject to linear equality constraints is described. The algorithm is based on a recently developed Newton barrier method for the unconstrained minimization of a sum of Euclidean norms (MSN). The linear equality constraints are handled using an exact L 1 penalty function which is made smooth in the same way as the Euclidean norms. It is shown...

In this paper, we empirically examine the use of a range of Minkowski norms for the clustering of real world data. We also investigate whether normalisation of the data prior to clustering affects the quality of the result. In a nearest neighbour search on raw real world data sets, fractional norms outperform the Euclidean and higher-order norms. However, when the data are normalised, the resul...

Motivated by the deterministic single exponential time algorithm of Micciancio and Voulgaris for solving the shortest and closest vector problem for the Euclidean norm, we study the geometry and complexity of Voronoi cells of lattices with respect to arbitrary norms. On the positive side, we show that for strictly convex and smooth norms the geometry of Voronoi cells of lattices in any dimensio...