Journal ArticleDOI
Each convex body inE 3 symmetric about a plane can be illuminated by 8 directions
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This article is published in Journal of Geometry.The article was published on 2000-11-01. It has received 12 citations till now. The article focuses on the topics: Convex hull & Convex set.read more
Citations
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Journal ArticleDOI
The illumination conjecture and its extensions
TL;DR: The paper surveys the state of the art of the Illumination Conjecture, which states that any d-dimensional convex body can be illuminated by at most $2^d$ light sources.
Book ChapterDOI
The geometry of homothetic covering and illumination
TL;DR: In this paper, the authors survey the recent advances in the area of illumination conjecture in discrete geometry, computational geometry, and geometric analysis, and describe two new approaches to make progress on the illumination problem.
Posted Content
Improved bounds for Hadwiger's covering problem via thin-shell estimates
TL;DR: The best known lower bound for the Hadwiger's covering problem is a sub-exponential lower bound of Ω(n 2 n ) for convex bodies with positive modulus of convexity as mentioned in this paper.
Book ChapterDOI
Flavors of Translative Coverings
TL;DR: In this paper, the authors survey results on the problem of covering the space of a convex body, or a body in it, by translates of the body with density around a constant.
DissertationDOI
Some Problems on Graphs and Arrangements of Convex Bodies
TL;DR: The covering index of a convex body is introduced as a measure of how economically the body can be covered by a relatively few smaller positive homothetic copies and the maximum contact numbers of totally separable translative packings of a smooth strictly convex domain are considered.
References
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Book ChapterDOI
Extremum Problems with Inequalities as Subsidiary Conditions
TL;DR: In this paper, an extension of Lagrange's multiplier rule to the case where the subsidiary conditions are inequalities instead of equations is considered, where only extrema of differentiable functions of a finite number of variables will be considered.
Book ChapterDOI
Aspects of Approximation of Convex Bodies
TL;DR: Approximation of convex bodies is frequently encountered in geometric convexity, discrete geometry, the theory of finite-dimensional normed spaces, in geometric algorithms and optimization, and in the realm of engineering as discussed by the authors.
Book ChapterDOI
The Local Theory of Normed Spaces and its Applications to Convexity
TL;DR: The local theory of normed spaces and its applications to convexity was introduced in this paper, where the main idea is to consider complex elements as points in a linear space and investigate the relation between such objects in analogy to the study of points, lines, and planes in geometry using an appropriate norm for quantitative estimates.
Journal ArticleDOI
Solution of Hadwiger's Covering Problem for Centrally Symmetric Convex Bodies in E3
Journal ArticleDOI
The problem of illumination of the boundary of a convex body by affine subspaces
TL;DR: The main result of as discussed by the authors is that any convex polytope of E d with affine symmetry can be illuminated by eight (d - 3)-dimensional affine subspaces.