Convexity in the Khalimsky Plane
-
Ezzeddine Bouassida
,
Riyadh Gargouri
Abstract
Let ℤ2 be equipped with the Khalimsky topology κ, it is a T0-Alexandroff topology which has some specific properties concerning continuity and connectivity. We define digital-arcs and the geodesics; this enables us to define D-convexity on the digital plane (ℤ2, κ). First, we prove a theorem dealing with the relationship between D-convexity and connectivity. The second result links together the convexity in ℝ2 and the D-convexity in ℤ2. For this purpose, we suggest the continuous digitization of the real line segment and thus prove that the digitization of a convex subset of ℝ2 is a D-convex subset of (ℤ2, κ).
References
[1] BOUACIDA, E.: The Jordan curve theorem in the Khalimsky plane, Appl. Gen. Topol. 9 (2008), 253-262.10.4995/agt.2008.1805Search in Google Scholar
[2] BOUACIDA, E.-JARBOUI, N.: Connectivity in A-spaces, JP J.Geom. Topol. 7 (2007), 309-320.Search in Google Scholar
[3] ECKHARDT, U.-LATECKI, L. J.: Topologies for the digital spaces Z2 and Z3, Comput. Vision Image Understanding 90 (2003), 295-312.10.1016/S1077-3142(03)00062-6Search in Google Scholar
[4] KHALIMSKY, E. D.: On topologies of generalized segments, Soviet Math. Dokl. 10 (1999), 1508-1511.Search in Google Scholar
[5] KHALIMSKY, E. D.-KOPPERMAN, R.-MEYER, P. R.: Boundaries in digital planes, J. Appl. Math. Stoch. Anal. 3 (1990), 27-55.10.1155/S1048953390000041Search in Google Scholar
[6] KHALIMSKY, E.-KOPPERMAN, R.-MEYER, P. R.: Computer graphics and connected topologies on finite ordered sets, Topology Appl. 36 (1990), 1-17.10.1016/0166-8641(90)90031-VSearch in Google Scholar
[7] KISELMAN, C. O.: Digital Jordan curve theorems. In: DGCI00. Proceedings of the 9th International Conference on Discrete Geometry for Computer Imagery, UK, 2000, Springer-Verlag, London, pp. 46-56.Search in Google Scholar
[8] KONG, T. Y.-KOPPERMAN, R.-MEYER, P. R.: A topological approach to digital topology, Amer. Math. Monthly 98 (1991), 902-917.10.1080/00029890.1991.12000810Search in Google Scholar
[9] MELIN, E.: Digital staight lines in the Khalimsky plane, Math. Scand. 96 (2005), 49-62.10.7146/math.scand.a-14944Search in Google Scholar
[10] SANG-EON HAN: Homotopy equivalence which is suitable for studying Khalimsky nD spaces, Topology Appl. 159 (2012), 1705-1714.10.1016/j.topol.2011.07.029Search in Google Scholar
[11] ŠLAPAL, J.: A digital analogue of the Jordan curve theorem, Discrete Appl. Math. 139 (2004), 231-251.10.1016/j.dam.2002.11.003Search in Google Scholar
[12] ŠLAPAL, J.: A quotient-universal digital topology, Theoret. Comput. Sci. 405 (2008), 164-175.10.1016/j.tcs.2008.06.035Search in Google Scholar
[13] ŠLAPAL, J.: Digital Jordan Curves, Topology Appl. 153 (2006), 3255-3264. 10.1016/j.topol.2005.10.011Search in Google Scholar
Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Radius, Diameter and the Degree Sequence of a Graph
- On Primary Ideals in Posets
- Characterization of the Set of Regular Elements in Ordered Semigroups
- Tame Automorphisms with Multidegrees in the Form of Arithmetic Progressions
- A Result Concerning Additive Mappings in Semiprime Rings
- Characterizing Jordan Derivations of Matrix Rings Through Zero Products
- Existence Results for Impulsive Nonlinear Fractional Differential Equations With Nonlocal Boundary Conditions
- The Radon-Nikodym Property and the Limit Average Range
- A Hake-Type Theorem for Integrals with Respect to Abstract Derivation Bases in the Riesz Space Setting
- On Booth Lemniscate and Hadamard Product of Analytic Functions
- Recursion Formulas for Srivastava Hypergeometric Functions
- Regularly Varying Solutions of Half-Linear Diffferential Equations with Retarded and Advanced Arguments
- Singular Degenerate Differential Operators and Applications
- The Interior Euler-Lagrange Operator in Field Theory
- On Selections of Set-Valued Maps Satisfying Some Inclusions in a Single Variable
- The Family F of Permutations of ℕ
- Summation Process of Positive Linear Operators in Two-Dimensional Weighted Spaces
- On Iλ-Statistical Convergence in Locally Solid Riesz Spaces
- Some Norm one Functions of the Volterra Operator
- Some Results on Absolute Retractivity of the Fixed Points Set of KS-Multifunctions
- Convexity in the Khalimsky Plane
- Natural Boundary Conditions in Geometric Calculus of Variations
- Exponential Inequalities for Bounded Random Variables
- Precise Rates in the Law of Iterated Logarithm for the Moment Convergence of φ-Mixing Sequences
- Second Order Riemannian Mechanics
- Further Remarks on an Order for Quantum Observables
Articles in the same Issue
- Radius, Diameter and the Degree Sequence of a Graph
- On Primary Ideals in Posets
- Characterization of the Set of Regular Elements in Ordered Semigroups
- Tame Automorphisms with Multidegrees in the Form of Arithmetic Progressions
- A Result Concerning Additive Mappings in Semiprime Rings
- Characterizing Jordan Derivations of Matrix Rings Through Zero Products
- Existence Results for Impulsive Nonlinear Fractional Differential Equations With Nonlocal Boundary Conditions
- The Radon-Nikodym Property and the Limit Average Range
- A Hake-Type Theorem for Integrals with Respect to Abstract Derivation Bases in the Riesz Space Setting
- On Booth Lemniscate and Hadamard Product of Analytic Functions
- Recursion Formulas for Srivastava Hypergeometric Functions
- Regularly Varying Solutions of Half-Linear Diffferential Equations with Retarded and Advanced Arguments
- Singular Degenerate Differential Operators and Applications
- The Interior Euler-Lagrange Operator in Field Theory
- On Selections of Set-Valued Maps Satisfying Some Inclusions in a Single Variable
- The Family F of Permutations of ℕ
- Summation Process of Positive Linear Operators in Two-Dimensional Weighted Spaces
- On Iλ-Statistical Convergence in Locally Solid Riesz Spaces
- Some Norm one Functions of the Volterra Operator
- Some Results on Absolute Retractivity of the Fixed Points Set of KS-Multifunctions
- Convexity in the Khalimsky Plane
- Natural Boundary Conditions in Geometric Calculus of Variations
- Exponential Inequalities for Bounded Random Variables
- Precise Rates in the Law of Iterated Logarithm for the Moment Convergence of φ-Mixing Sequences
- Second Order Riemannian Mechanics
- Further Remarks on an Order for Quantum Observables
