Research

My main research interest is Minkowski geometry, which consists in the geometry of real finite dimensional vector spaces endowed with a norm. There are two aspects of the field that, in my opinion, make it so beautiful and important: first, it is a very “geometric flavored” area subject. Second, Minkowski geometry is related (or can be related) to a lot of areas of Mathematics, such as functional analysis, convex geometry, distance geometry and Finsler geometry. This relation occurs in two directions: one can use Minkowski geometry to achive a better understanding about some of this topics, or also study Minkowski geometry with tools and methods inherited from them. Click on “Publications” to see the papers related to each of these topics.

I’m currently working on the theory of differential geometry in Minkowski spaces. We want to study curves and surfaces immersed in finite dimensional vector spaces endowed with a norm. The idea is to develop an analogue theory, but using general norms instead of norms derived from an inner product. There are some papers in the literature dealing with this, but they are sparse, and the topic was never given a systematic treatment.

I’m also interested in the theory of geometric constants in normed spaces. Roughly speaking, geometric constants quantify “distortions” in a space endowed with a norm. I have defined and studied constants that can measure how far a normed plane is from being Radon or Euclidean, and also how different are orthogonality types in a normed plane.

Further on, I have studied angle concepts and trigonometric functions in Minkowski spaces. A space endowed with a norm does not have a “natural” definition of angle, but there are a plenty of analogous definitions that make sense. Moreover, one can define analogous to trigonometric functions and study their properties. As one may expect, these functions have a lot of applications. For example, they can be used to obtain characterizations of Radon and Euclidean planes, and also to define geometric constants.

As mentioned, the first topic is holding my attention for now. The other both are subjects that I have studied during my last doctoral years, but I still have interest in them. I’m also interested in other subtopics of Minkowski geometry that I have not dealt with (but intend to), such as concepts of area and volume, the geometry of the sphere (in the sense of Schaffer’s book) and discrete geometry in normed spaces.

I’m also interested in the geometry of convex distance functions in finite-dimensional vector spaces (also known by gauges). Less related to Minkowski geometry, I’ve been recently studying analysis on metric-measure spaces, with some topological dynamics involved.