A subset X of the integer grid 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.[1]
Definitions
Let X be a subset of .
Denote by ch(X) the convex hull of X. Note that ch(X) is a subset of , since it contains all the real points that are convex combinations of the integer points in X.
For any point y in , denote near(y) := {z in | |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 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.
In contrast, the set Y = { (0,0), (1,0), (2,0), (1,1), (2,1) } is integrally convex.
Properties
Iimura, Murota and Tamura[3] have shown the following property of integrally convex set.
Let 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 .
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.
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. ISBN978-3-540-36926-4.
This article uses material from the Wikipedia article Integrally-convex_set, and is written by contributors.
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