0/1-polytope

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A 0/1-polytope is a convex polytope generated by the convex hull of a subset of $d$ coordinates value 0 or 1, ${0,1} d$ .^[1] The full domain is the unit hypercube with cut hyperplanes passing through these coordinates.^[2] A $d$ -polytope requires at least $d + 1$ vertices, and can't be all in the same hyperplanes.

n-simplex polytopes for example can be generated $n + 1$ vertices, using the origin, and one vertex along each primary axis, $(1,0....)$ , etc. Every simple 0/1-polytope is a Cartesian product of 0/1 simplexes.^[3]

References

^ Ziegler, Günter M. (2000). "Lectures on 0/1-polytopes". Polytopes—combinatorics and computation (Oberwolfach, 1997). DMV Sem. Vol. 29. Basel: Birkhäuser. pp. 1–41. ISBN 3-7643-6351-7. MR 1785291.
^ Grünbaum, Branko (2003). "4.9. Additional notes and comments". Convex Polytopes. Springer. p. 69a.
^ Kaibel, Volker; Wolff, Martin (2000). "Simple 0/1-polytopes". European Journal of Combinatorics. 21 (1): 139–144. doi:10.1006/eujc.1999.0328. MR 1737334.

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[ziegler-1] Ziegler, Günter M. (2000). "Lectures on 0/1-polytopes". Polytopes—combinatorics and computation (Oberwolfach, 1997). DMV Sem. Vol. 29. Basel: Birkhäuser. pp. 1–41. ISBN 3-7643-6351-7. MR 1785291.

[grunbaum-2] Grünbaum, Branko (2003). "4.9. Additional notes and comments". Convex Polytopes. Springer. p. 69a.

[kw-3] Kaibel, Volker; Wolff, Martin (2000). "Simple 0/1-polytopes". European Journal of Combinatorics. 21 (1): 139–144. doi:10.1006/eujc.1999.0328. MR 1737334.

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