Discrete Geometry in Minkowski Spaces

AbstractA remarkable family of discrete sets which has recently attracted the attention of the discrete geometry community is the family of convex polyominoes, that are the discrete counterpart of Euclidean convex sets, and combine the constraints of convexity and connectedness. In this paper we study the problem of their reconstruction from orthogonal projections, relying on the approach defined by Barcucci et al. (Theor Comput Sci 155(2):321–347, 1996). In particular, during the reconstruction process it may be necessary to expand a convex subset of the interior part of the polyomino, say the polyomino kernel, by adding points at specific positions of its contour, without losing its convexity. To reach this goal we consider convexity in terms of certain combinatorial properties of the boundary word encoding the polyomino. So, we first show some conditions that allow us to extend the kernel maintaining the convexity. Then, we provide examples where the addition of one or two points causes a loss of convexity, which can be restored by adding other points, whose number and positions cannot be determined a priori.

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Masslessness and light-cone propagation in 3+2 de Sitter and 2+1 Minkowski spaces

Physical Review D ◽

10.1103/physrevd.33.415 ◽

1986 ◽

Vol 33 (2) ◽

pp. 415-420 ◽

Cited By ~ 14

Author(s):

M. Flato ◽

C. Fronsdal ◽

J. P. Gazeau

Keyword(s):

Light Cone ◽

De Sitter ◽

Minkowski Spaces

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Volume decay and concentration of high-dimensional Euclidean balls – a PDE and variational perspective

Analysis ◽

10.1515/anly-2020-0035 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Siran Li

Keyword(s):

Banach Space ◽

Discrete Geometry ◽

Thin Shell ◽

Euclidean Geometry ◽

Convex Geometry ◽

Regularity Theory ◽

Elliptic Partial Differential Equations ◽

High Dimensional ◽

Space Theory ◽

Elementary Calculus

AbstractIt is a well-known fact – which can be shown by elementary calculus – that the volume of the unit ball in \mathbb{R}^{n} decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as n\nearrow\infty. Many rigorous proofs and heuristic arguments are provided for this fact from different viewpoints, including Euclidean geometry, convex geometry, Banach space theory, combinatorics, probability, discrete geometry, etc. In this note, we give yet another two proofs via the regularity theory of elliptic partial differential equations and calculus of variations.

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