The least dense lattice packing of two-dimensional convex bodies

1965 ◽  
Vol 18 (1-2) ◽  
pp. 339-343 ◽  
Author(s):  
R. Courant
Keyword(s):  
Convex Bodies ◽  
Lattice Packing ◽  
Two Dimensional ◽  
Dense Lattice ◽  
Dense Lattice Packing

Convex bodies with equiframed two-dimensional sections

2007 ◽  
Vol 88 (2) ◽  
pp. 181-192 ◽  
Author(s):  
Nico Düvelmeyer
Keyword(s):  
Convex Bodies ◽  
Two Dimensional

Extremal Kähler–Einstein Metric for Two-Dimensional Convex Bodies

2018 ◽  
Vol 29 (3) ◽  
pp. 2347-2373
Author(s):  
Bo’az Klartag ◽  
Alexander V. Kolesnikov
Keyword(s):  
Convex Bodies ◽  
Einstein Metric ◽  
Two Dimensional

scholarly journals SOLUTION OF THE TWO-DIMENSIONAL WAVE DIFFRACTION PROBLEM ON FRACTAL BODIES BY THE PATTERN EQUATIONS METHOD

T-Comm ◽  
2021 ◽  
Vol 15 (9) ◽  
pp. 4-10
Author(s):  
Aleksey S. Davydov ◽  
◽  
Dmitry B. Demin ◽  
Dmitry V. Krysanov ◽  
◽  
...  
Keyword(s):  
Wave Diffraction ◽  
Diffraction Problem ◽  
Convex Bodies ◽  
Numerical Algorithms ◽  
Dirichlet Condition ◽  
Infinite Cylinder ◽  
Optical Theorem ◽  
Two Dimensional ◽  
Koch Snowflake ◽  
Dimensional Wave

The solution of the two-dimensional wave diffraction problem for infinite cylinder of complex cross-section was considered by using the pattern equations method (PEM). A triangle and a Koch snowflake of first iteration were chosen as the geometry of the cross-sections of the cylinder. The numerical algorithms of the PEM for a single scatterer and for a group of bodies with the Dirichlet condition on their boundary are briefly presented, and the results of numerical calculations of the scattering characteristics for the above geometries are obtained using the PEM and the method of continued boundary conditions (MCBC). To check the convergence of the numerical algorithm in both methods, the optical theorem was used. The limits of applicability of the PEM for fractal scatterers are established. It is shown that for all convex bodies the algorithm of the PEM is sufficiently stable and allows obtaining calculation results with an accuracy acceptable in practice. In the case of a non-convex body, namely, a Koch snowflake, the algorithm of the PEM for a single scatterer turns out to be unstable and the acceptable accuracy can be obtained only if this geometry is considered as a group of bodies composed of convex geometries (for example, triangles).

scholarly journals An Upper Bound for Volumes of Convex Bodies

1969 ◽  
Vol 9 (3-4) ◽  
pp. 503-510
Author(s):  
William J. Firey
Keyword(s):  
Convex Body ◽  
Orthogonal Projection ◽  
Upper Bound ◽  
Linear Subspace ◽  
Dimensional Space ◽  
Convex Bodies ◽  
Maximum Length ◽  
Two Dimensional ◽  
Image Position

Consider a non-degenerate convex body K in a Euclidean (n + 1)-dimensional space of points (x, z) = (x1,…, xn, z) where n ≧2. Denote by μ the maximum length of segments in K which are parallel to the z-axis, and let Aj, signify the area (two dimensional volume) of the orthogonal projection of K onto the linear subspace spanned by the z- and xj,-axes. We shall prove that the volume V(K) of K satisfies After this, some applications of (1) are discussed.

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