An integrally convex set is the discrete geometry analogue of the concept of convex set in geometry.
A subset X of the integer grid Z n {\displaystyle \mathbb {Z} ^{n}} is integrally convex if any point y in the convex hull of X can be expressed as a convex combination of the points of X that are "near" y, where "near" means that the distance between each two coordinates is less than 1.
Definitions
Let X be a subset of Z n {\displaystyle \mathbb {Z} ^{n}} .
Denote by ch(X) the convex hull of X. Note that ch(X) is a subset of R n {\displaystyle \mathbb {R} ^{n}} , since it contains all the real points that are convex combinations of the integer points in X.
For any point y in R n {\displaystyle \mathbb {R} ^{n}} , denote near(y) := {z in Z n {\displaystyle \mathbb {Z} ^{n}} | |zi - yi| < 1 for all i in {1,...,n} }. These are the integer points that are considered "nearby" to the real point y.
A subset X of Z n {\displaystyle \mathbb {Z} ^{n}} is called integrally convex if every point y in ch(X) is also in ch(X ∩ near(y)).2
Example
Let n = 2 and let X = { (0,0), (1,0), (2,0), (2,1) }. Its convex hull ch(X) contains, for example, the point y = (1.2, 0.5).
The integer points nearby y are near(y) = {(1,0), (2,0), (1,1), (2,1) }. So X ∩ near(y) = {(1,0), (2,0), (2,1)}. But y is not in ch(X ∩ near(y)). See image at the right.
Therefore X is not integrally convex.3
In contrast, the set Y = { (0,0), (1,0), (2,0), (1,1), (2,1) } is integrally convex.
Properties
Iimura, Murota and Tamura4 have shown the following property of integrally convex set.
Let X ⊂ Z n {\displaystyle X\subset \mathbb {Z} ^{n}} be a finite integrally convex set. There exists a triangulation of ch(X) that is integral, i.e.:
- The vertices of the triangulation are the vertices of X;
- The vertices of every simplex of the triangulation lie in the same "cell" (hypercube of side-length 1) of the integer grid Z n {\displaystyle \mathbb {Z} ^{n}} .
The example set X is not integrally convex, and indeed ch(X) does not admit an integral triangulation: every triangulation of ch(X), either has to add vertices not in X, or has to include simplices that are not contained in a single cell.
In contrast, the set Y = { (0,0), (1,0), (2,0), (1,1), (2,1) } is integrally convex, and indeed admits an integral triangulation, e.g. with the three simplices {(0,0),(1,0),(1,1)} and {(1,0),(2,0),(2,1)} and {(1,0),(1,1),(2,1)}. See image at the right.
References
Yang, Zaifu (2009-12-01). "Discrete fixed point analysis and its applications". Journal of Fixed Point Theory and Applications. 6 (2): 351–371. doi:10.1007/s11784-009-0130-9. ISSN 1661-7746. S2CID 122640338. /wiki/Doi_(identifier) ↩
Chen, Xi; Deng, Xiaotie (2006). "A Simplicial Approach for Discrete Fixed Point Theorems". In Chen, Danny Z.; Lee, D. T. (eds.). Computing and Combinatorics. Lecture Notes in Computer Science. Vol. 4112. Berlin, Heidelberg: Springer. pp. 3–12. doi:10.1007/11809678_3. ISBN 978-3-540-36926-4. 978-3-540-36926-4 ↩
Yang, Zaifu (2009-12-01). "Discrete fixed point analysis and its applications". Journal of Fixed Point Theory and Applications. 6 (2): 351–371. doi:10.1007/s11784-009-0130-9. ISSN 1661-7746. S2CID 122640338. /wiki/Doi_(identifier) ↩
Iimura, Takuya; Murota, Kazuo; Tamura, Akihisa (2005-12-01). "Discrete fixed point theorem reconsidered" (PDF). Journal of Mathematical Economics. 41 (8): 1030–1036. doi:10.1016/j.jmateco.2005.03.001. ISSN 0304-4068. http://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1449.pdf ↩