Computational geometry: an introduction
Computational geometry: an introduction
Combinatorial optimization: algorithms and complexity
Combinatorial optimization: algorithms and complexity
On the maximal number of edges of convex digital polygons included into an m × m-grid
Journal of Combinatorial Theory Series A
Real data—integer solution problems with the Blum-Shub-Smale computational model
Journal of Complexity
Polyhedra generation from lattice points
DCGA '96 Proceedings of the 6th International Workshop on Discrete Geometry for Computer Imagery
Polyhedrization of discrete convex volumes
ISVC'06 Proceedings of the Second international conference on Advances in Visual Computing - Volume Part I
Maximal digital straight segments and convergence of discrete geometric estimators
SCIA'05 Proceedings of the 14th Scandinavian conference on Image Analysis
DGCI'06 Proceedings of the 13th international conference on Discrete Geometry for Computer Imagery
Computational aspects of digital plane and hyperplane recognition
IWCIA'06 Proceedings of the 11th international conference on Combinatorial Image Analysis
Some theoretical challenges in digital geometry: A perspective
Discrete Applied Mathematics
On the Convex Hull of the Integer Points in a Bi-circular Region
IWCIA '09 Proceedings of the 13th International Workshop on Combinatorial Image Analysis
Determining Digital Circularity Using Integer Intervals
Journal of Mathematical Imaging and Vision
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Let B^n be a hyperball in R^n, n=2, and denote B"Z^n=B^n@?Z^n. Define polyhedral facet complexity of B"Z^n as FC(B"Z^n)=min"P{f"n"-"1(P)} where P is an enclosing polyhedron for B"Z^n (i.e., P"Z=P@?Z^n=B"Z^n) and f"n"-"1(P) is the number of the (n-1)-facets of P. Analogously, define polyhedral vertex complexity of B"Z^n as VC(B"Z^n)=min"P{f"0(P)} where P is an enclosing polyhedron for B"Z^n and f"0(P) is the number of the 0-facets (vertices) of P. Upper bounds for FC(B"Z^n) follow from a well-known bound for the number of facets and vertices of the convex hull of B"Z^n [I. Barany, D.G. Larman, The convex hull of the integer points in a large ball, Math. Ann. 312 (1998) 167-181]. In this note we provide the first nontrivial lower bounds on FC(B"Z^n) and VC(B"Z^n).