Ákos G. Horváth

Bio: Ákos G. Horváth is an academic researcher from Budapest University of Technology and Economics. The author has contributed to research in topics: Unit sphere & Convex hull. The author has an hindex of 11, co-authored 84 publications receiving 517 citations.

On Bisectors in Minkowski Normed Spaces

Abstract: We discuss the concept of the bisector of a segment in a Minkowski normed n-space, and prove that if the unit ball K of the space is strictly convex then all bisectors are topological images of a hyperplane of the embedding Euclidean n-space. The converse statement is not true. We give an example in the three-space showing that all bisectors are topological planes, however K contains segments on its boundary. Strict convexity ensures the normality of Dirichlet-Voronoi-type K-subdivision of any point lattice.

Angles in normed spaces

Abstract: The concepts of angle, angle functions, and the question how to measure angles present old and well-established mathematical topics referring to the Euclidean space, and there exist also various extensions to non-Euclidean spaces of different types. In particular, it is very interesting to investigate or to combine (geometric) properties of possible concepts of angle functions and angle measures in finite-dimensional real Banach spaces (= Minkowski spaces). However, going into this direction one will observe that there is no monograph or survey reflecting the complete picture of the existing literature on such concepts in a satisfying manner. We try to close this gap. In this expository paper (containing also new results, and new proofs of known results) the reader will get a comprehensive overview of this field, including further related aspects, as well. For example, angular bisectors, their applications, and angle types which preserve certain kinds of orthogonality are discussed. The latter aspect yields, of course, an interesting link to the large variety of orthogonality types in such spaces.

Maximum volume polytopes inscribed in the unit sphere

Abstract: In this paper we investigate the problem of finding the maximum volume polytopes, inscribed in the unit sphere of the d-dimensional Euclidean space, with a given number of vertices. We solve this problem for polytopes with \(d+2\) vertices in every dimension, and for polytopes with \(d+3\) vertices in odd dimensions. For polytopes with \(d+3\) vertices in even dimensions we give a partial solution.

A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations

Abstract: We give deterministic ~O(22n+o(n))-time algorithms to solve all the most important computational problems on point lattices in NP, including the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), and Shortest Independent Vectors Problem (SIVP). This improves the nO(n) running time of the best previously known algorithms for CVP (Kannan, Math. Operation Research 12(3):415--440, 1987) and SIVP (Micciancio, Proc. of SODA, 2008), and gives a deterministic and asymptotically faster alternative to the 2O(n)-time (and space) randomized algorithm for SVP of (Ajtai, Kumar and Sivakumar, STOC 2001). The core of our algorithm is a new method to solve the closest vector problem with preprocessing (CVPP) that uses the Voronoi cell of the lattice (described as intersection of half-spaces) as the result of the preprocessing function. In the process, we also give algorithms for several other lattice problems, including computing the kissing number of a lattice, and computing the set of all Voronoi relevant vectors. All our algorithms are deterministic, and have 2O(n) time and space complexity.

On Uniformly Convex Spaces

Abstract: (b) Show that if X is uniformly convex and (xn) ⊂ X such that lim n→∞ ‖xn‖ = 1 and lim n,m→∞ ‖xn + xm‖ = 2, then (xn) is convergent. Proceed as follows. Use the definition to prove first that if all terms in the sequence have unit length, then the sequence is Cauchy. In the general case, let yn = xn/‖xn‖. Prove that lim n,m→∞ ‖yn + ym‖ = 2. Then, combine this with what you have shown already to prove that (xn) must be Cauchy. (c) Prove that if X is uniformly convex and S ⊂ X is closed and convex, then S contains a unique element of minimal norm, that is, a unique s0 ∈ S such that inf s∈S ‖s‖ = ‖s0‖. Proceed as follows. Let d = inf s∈S ‖s‖. Now, use part (b) to construct a

On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces

Abstract: We survey mainly recent results on the two most important orthogonality types in normed linear spaces, namely on Birkhoff orthogonality and on isosceles (or James) orthogonality. We lay special emphasis on their fundamental properties, on their differences and connections, and on geometric results and problems inspired by the respective theoretical framework. At the beginning we also present other interesting types of orthogonality. This survey can also be taken as an update of existing related representations.