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Russia Moscow
Section Mathematics
Title On a cube and subspace projections
Author(-s) Boykov A.A.a, Seliverstov A.V.b
Affiliations MIREA - Russian Technological Universitya, Institute for Information Transmission Problems, Russian Academy of Sciencesb
Abstract We consider the arrangement of vertices of a unit multidimensional cube, an affine subspace, and its orthogonal projections onto coordinate subspaces. Upper and lower bounds on the subspace dimension are given under which some orthogonal projection always preserves the incidence relation between the subspace and cube vertices. Some oblique projections are also considered. Moreover, a brief review of the history of the development of multidimensional descriptive geometry is given. Analytic and synthetic methods in geometry diverged since the 17th century. Although both synthesis and analysis are tangled, from this time forth many geometers as well as engineers keep up a nice distinction. One can find references to the idea of higher-dimensional spaces in the 18th-century works, but proper development has been since the middle of the 19th century. Soon such works have appeared in Russian. Next, mathematicians generalized their theories to many dimensions. Our new results are obtained by both analytic and synthetic methods. They illustrate the complexity of pseudo-Boolean programming problems because reducing the problem dimension by orthogonal projection meets obstacles in the worst case.
Keywords multidimensional cube, affine subspace, projection, discrete optimization, history of mathematics
UDC 514.142
MSC 51A15, 51N05
DOI 10.35634/vm230302
Received 10 January 2023
Language Russian
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