Related papers: Geometric valuation theory
The geometric mean of two matrices is considered and analyzed from a computational viewpoint. Some useful theoretical properties are derived and an analysis of the conditioning is performed. Several numerical algorithms based on different…
We recall the space of seminorms discussed by Payne in \cite{P} and define a slight modification, the space of graded valuations. After explaining how these spaces relate to tropical geometry, we describe examples of graded valuations which…
Convexity and convex functions play an important role in theoretical physics. To initiate a study of the possible uses of convex functions in General Relativity, we discuss the consequences of a spacetime $(M,g_{\mu \nu})$ or an initial…
Convexity and convex functions play an important role in theoretical physics. To initiate a study of the possible uses of convex functions in General Relativity, we discuss the consequences of a spacetime $(M,g_{\mu \nu})$ or an initial…
We give a characterization of smooth, rotation and dually epi-translation invariant valuations and use this result to obtain a new proof of the Hadwiger theorem on convex functions. We also give a description of the construction of the…
A complete classification of all zonal, continuous, and translation invariant valuations on convex bodies is established. The valuations obtained are expressed as principal value integrals with respect to the area measures. The convergence…
We introduce the concepts of a pair of valuations and a good generating set and show how they can be used to prove geometric properties of soluble groups.
We review the definition of geometric quantization, which begins with defining a mathematical framework for the algebra of observables that holds equally well for classical and quantum mechanics. We then discuss prequantization, and go into…
This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to…
We introduce vectorial and topological continuities for functions defined on vector metric spaces and illustrate spaces of such functions. Also, we describe some fundamental classes of vector valued functions and extension theorems.
In this paper, we explain how some basic facts about valuation can help clarify many questions about divisibility in integral domains.
Very recently J. Kotrbaty has proven general inequalities for translation invariant smooth valuations formally analogous to the Hodge- Riemann bilinear relations in the Kahler geometry. The goal of this note is to apply Kotrbaty's theorem…
We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of…
Valuations on the space of finite-valued convex functions on $\mathbb{C}^n$ that are continuous, dually epi-translation invariant, as well as $\mathrm{U}(n)$-invariant are completely classified. It is shown that the space of these…
We study the space of generalized translation invariant valuations on a finite-dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations. Our main theorem is that…
This note is an invitation to the theory of geometric functions. The foundation techniques and some of the developments in the field are explained with the mindset that the audience is principally young researchers wishing to understand…
A classification of SL$(n)$ invariant valuations on the space of convex polytopes in $R^n$ without any continuity assumptions is established. A corresponding result is obtained on the space of convex polytopes in $R^n$ that contain the…
The goal of this article is to present a survey of the recent theory of plurisubharmonic functions of quaternionic variables, and its applications to theory of valuations on convex sets and HKT-geometry (HyperK\"ahler with Torsion). The…
The goal of inversion is to estimate the model which generates the data of observations with a specific modeling equation. One general approach to inversion is to use optimization methods which are algebraic in nature to define an objective…
The article presents a new method of integration of functions with values in Banach spaces. This integral and related notions prove to be a useful tool in the study of Banach space geomtry.