Definition
Informally, if P is any polyhedron or polytope, and tP is the polytope formed by expanding P by a factor of t in each dimension, then L(int P, t) is the number of integer lattice points in tP.
More formally, consider a lattice L in Euclidean space Rn and a d-dimensional polytope P in Rn, and assume that all the vertices of the polytope are points of the lattice. (A common example is L = Zn and a polytope with all its vertex coordinates being integers.) For any positive integer t, let tP be the t-fold dilation of P (the polytope formed by multiplying each vertex coordinate, in a basis for the lattice, by a factor of t), and let
be the number of lattice points contained in tP. Ehrhart showed in 1962 that L is a rational polynomial of degree d in t, i.e. there exist rational numbers a0,...,ad such that:
- L(P, t) = adtd + ad−1td−1 + … + a0 for all positive integers t.
The Ehrhart polynomial of the interior of a closed convex polytope P can be computed as:
- L(int P, t) = (−1)n L(P, −t).
Read more about this topic: Ehrhart Polynomial
Famous quotes containing the word definition:
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)