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There are many problems and configurations in Euclidean geometry that were never extended to the framework of (normed or) finite dimensional real Banach spaces, although their original versions are inspiring for this type of generalization,…

The geometry of 2D Minkowski spacetime $\mathbb{R}^{1,1}$ (or Minkowski plane) is similar but fundamentally different from the more familiar Euclidean plane geometry. This note gives an elementary discussion on some basic properties of a…

In this paper we review the known facts on isometries of Minkowski geometries and prove some new results on them. We give the normal forms of two special classes of operators and also characterize the isometry group of Minkowski $3$-spaces…

A new way to define the notion of $\C$-orthocenter will be displayed by studying some propierties of four points in the plane which allows to extend the notion of Euler's line, the Six Point Circles and the three-circles theorem, for normed…

Since the end of the 19th century, and after the works of F. Klein and H. Poincar\'e, it is well known that models of elliptic geometry and hyperbolic geometry can be given using projective geometry, and that Euclidean geometry can be seen…

We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have…

We review part of the classical theory of curves and surfaces in $3$-dimensional Lorentz-Minkowski space. We focus in spacelike surfaces with constant mean curvature pointing the differences and similarities with the Euclidean space.

This paper considers the radii functionals (circumradius, inradius, and diameter) as well as the Minkowski asymmetry for general (possibly non-symmetric) gauge bodies. A generalization of the concentricity inequality (which states that the…

We prove three facts about intrinsic geometry of surfaces in a normed (Minkowski) space. When put together, these facts demonstrate a rather intriguing picture. We show that (1) geodesics on saddle surfaces (in a space of any dimension)…

The inscribed angle theorem, a famous result about the angle subtended by a chord within a circle, is well known and commonly taught in school curricula. In this paper, we present a generalisation of this result (and other related circle…

Penrose's Spin Geometry Theorem is extended further, from $SU(2)$ and $E(3)$ (Euclidean) to $E(1,3)$ (Poincar\'e) invariant elementary quantum mechanical systems. The Lorentzian spatial distance between any two non-parallel timelike…

This article derives closed-form parametric formulas for the Minkowski sums of convex bodies in d-dimensional Euclidean space with boundaries that are smooth and have all positive sectional curvatures at every point. Under these conditions,…

We first define a complex angle between two oriented spacelike planes in 4-dimensional Minkowski space, and then study the constant angle surfaces in that space, i.e. the oriented spacelike surfaces whose tangent planes form a constant…

We consider a generalized angle in complex normed vector spaces. Its definition corresponds to the definition of the well known Euclidean angle in real inner product spaces. Not surprisingly it yields complex values as `angles'. This…

We investigate the geometric properties of simplices in Euclidean d-dimensional space for which two or more of the analogues of the classical triangle centers (including the centroid, circumcenter, incenter, orthocenter or Monge point, and…

The normal map given by Birkhoff orthogonality yields extensions of principal, Gaussian and mean curvatures to surfaces immersed in three-dimensional spaces whose geometry is given by an arbitrary norm and which are also called Minkowski…

We define the simplest log-euclidean geometry. This geometry exposes a difficulty hidden in Hilbert's list of axioms presented in his "Grundlagen der Geometrie". The list of axioms appears to be incomplete if the foundations of geometry are…

This paper concerns the \textbf{abstract geometry of numbers}: namely the pursuit of certain aspects of geometry of numbers over a suitable class of normed domains. (The standard geometry of numbers is then viewed as geometry of numbers…

We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine…

The concept of angle, angle functions, and the question how to measure angles present old and well-established mathematical topics referring to Euclidean space, and there exist also various extensions to non-Euclidean spaces of different…