The eight lines connecting the vertices of the two cubes in this case represent a single direction in the "unseen" fourth dimension. This can be seen in the accompanying animation whenever it shows a smaller inner cube inside a larger outer cube. The simplest form of Hinton's method is to draw two ordinary 3D cubes in 2D space, one encompassing the other, separated by an "unseen" distance, and then draw lines between their equivalent vertices. In 1880, Charles Howard Hinton popularized these insights in an essay titled " What is the Fourth Dimension?", which explained the concept of a " four-dimensional cube" with a step-by-step generalization of the properties of lines, squares, and cubes. The idea of adding a fourth dimension began with Jean le Rond d'Alembert's "Dimensions" being published in 1754, was followed by Joseph-Louis Lagrange in the mid-1700s, and culminated in a precise formalization of the concept in 1854 by Bernhard Riemann. For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled x, y, and z). Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world. doi:10.1016/j.topol.2011.05.A four-dimensional space ( 4D) is a mathematical extension of the concept of three-dimensional space (3D). ![]() ![]() Our aim is to demonstrate that the example presented in * Corresponding author. However, the extension properties of these hyperspaces in the asymptotic category remained unknown. In the case of compact metric spaces as well as in the case of compact spaces of weight ω 1, the hyperspaces of compact convex subsets of probability measures are known to be absolute extensors. Note that these hyperspaces play an important role in the decision theory, mathematical economics and finance, in particular, in the maximum (maxmin) expected utility theory (cf. In the present paper we deal with the hyperspaces of compact convex subsets of probability measures. This leads to an open problem of searching functorial constructions that preserve the class of absolute extensors in the asymptotic categories. This provided a negative answer to a question formulated by Dranishnikov, in connection with existence of the homotopy extension theorem in this category. It was proved in that in general, the space of probability measures of a metric space is not an absolute extensor for the Dranishnikov category. ![]() Among the two categories widely used in asymptotic category, the Dranishnikov and the Roe categories (see the definition below), it turns out that it is the Dranishnikov category (the category of proper metric spaces and the asymptotically Lipschitz maps) in which a richer extensor theory can be developed. In asymptotic topology, the absolute extensors are used in constructing the homotopy theory and the asymptotic dimension theory. Introduction The notion of absolute extensor plays an important role in different branches of mathematics. Keywords: Compact convex set Probability measure Asymptotically zero-dimensional space Absolute extensor 1. In this paper we provide an example of an asymptotically zero-dimensional space (in the sense of Gromov) whose space of compact convex subsets of probability measures is not an absolute extensor in the asymptotic category in the sense of Dranishnikov. 2964, Ljubljana, 1001, Slovenia b Department of Mechanics and Mathematics, Lviv National University, Universytetska 1, 79000 Lviv, Ukraine c Institute of Mathematics, University of Rzeszów, Rzeszów, Poland article info abstract Article history: Received 18 February 2010 Received in revised form 18 April 2011 Accepted MSC: primary 46E27, 46E30 secondary 54C55, 54E35 The objects of the Dranishnikov asymptotic category are proper metric spaces and the morphisms are asymptotically Lipschitz maps. Topology and its Applications 158 (2011) 1571–1574 Contents lists available at ScienceDirect Topology and its Applications Convex hyperspaces of probability measures and extensors in the asymptotic category Dušan Repovš a,∗, Mykhailo Zarichnyi b,c a Faculty of Mathematics and Physics, and Faculty of Education, University of Ljubljana, P.O.B.
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