In this second part of a series of surveys on the geometry of finite dimensional Banach spaces (Minkowski spaces) we discuss results that refer to the following three topics: bodies of constant Minkowski width, generalized convexity notions that are important for Minkowski spaces, and bisectors as well as Voronoi diagrams in Minkowski spaces.

Standard lattice-space formulations of quartic self-coupled Euclidean scalar quantum fields become trivial in the continuum limit for sufficiently high space-time dimensions, and in particular the moment generating functional for space-time smeared fields becomes a Gaussian appropriate to that of a (possibly generalized) free field. For sharp-time fields this fact implies that the ground-state ...

in this paper, the time-like hyperruled surfaces in the minkowski 4-space and their algebraicinvariants are worked. also some characteristic results are found about these algebraic invariants.

Since its creation by Brunn and Minkowski, what has become known as the Brunn Minkowski theory has provided powerful machinery to solve a broad variety of inverse problems with stereological data. The machinery of the Brunn Minkowski theory includes mixed volumes (of Minkowski), symmetrization techniques (such as those of Steiner and Blaschke), isoperimetric inequalities (such as the Brunn Mink...

– The Brunn-Minkowski theory is a central part of convex geometry. At its foundation lies the Minkowski addition of convex bodies which led to the definition of mixed volume of convex bodies and to various notions and inequalities in convex geometry. Its origins were in Minkowski’s joining his notion of mixed volumes with the Brunn-Minkowski inequality, which dated back to 1887. Since then it h...

The Minkowski weighted K-means (MWK-means) is a recently developed clustering algorithm capable of computing feature weights. The cluster-specific weights in MWK-means follow the intuitive idea that a feature with low variance should have a greater weight than a feature with high variance. The final clustering found by this algorithm depends on the selection of the Minkowski distance exponent. ...

Minkowski Weighted K-Means is a variant of K-Means set in the Minkowski space, automatically computing weights for features at each cluster. As a variant of K-Means, its accuracy heavily depends on the initial centroids fed to it. In this paper we discuss our experiments comparing six initializations, random and five other initializations in the Minkowski space, in terms of their accuracy, proc...

The Minkowski distance is a distance measure that generalizes a wide range of other distances such as the Hamming and the Euclidean distance. In this paper, we develop a generalization of the Minkowski distance by using the induced ordered weighted averaging (IOWA) operator. We will call it the induced Minkowski OWA distance (IMOWAD). Then, we are able to obtain a wider range of distance measur...