– 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...

We introduce a new operation between nonnegative integrable functions on R n $\mathbb {R}^n$ , that we call geometric combination; it is obtained via mass transportation approach, playing with inverse distribution functions. The main feature of this the Lebesgue integral combination equals mean two separate integrals; as natural consequence, derive functional inequality Prékopa–Leindler type. W...

We study the Minkowski length L(P ) of a lattice polytope P , which is defined to be the largest number of non-trivial primitive segments whose Minkowski sum lies in P . The Minkowski length represents the largest possible number of factors in a factorization of polynomials with exponent vectors in P , and shows up in lower bounds for the minimum distance of toric codes. In this paper we give a...

Minkowski Pythagorean hodograph curves are polynomial curves with polynomial speed, measured with respect to Minkowski norm. Curves of this special class are particularly well suited for representing medial axis transforms of planar domains. In the present paper we generalize this polynomial class to a rational class of curves in Minkowski 3-space. We show that any rational Minkowski Pythagorea...

In this paper we parallelly build up the theories of normed linear spaces and of linear spaces with indefinite metric, called also Minkowski spaces for finite dimensions in the literature. In the first part of this paper we collect the common properties of the semi-and indefinite-inner-products and define the semi-indefinite-inner-product and the corresponding structure, the semi-indefinite-inn...

Log-Brunn-Minkowski inequality was conjectured by Boröczky, Lutwak, Yang and Zhang [7], and it states that a certain strengthening of the classical Brunn-Minkowski inequality is admissible in the case of symmetric convex sets. It was recently shown by Nayar, Zvavitch, the second and the third authors [27], that Log-Brunn-Minkowski inequality implies a certain dimensional Brunn-Minkowski inequal...

An effective Minkowski spacetime having no deformation in formal is established by introducing a well-defined proper time from the κ-deformed Minkowski spacetime related to the standard basis. The κ-Minkowski spacetime connects with a special curved spacetime through this effective spacetime. In the framework of such an effective Minkowski spacetime as a mapped coordinate system of the κ-deform...