Notes on convex sets, polytopes, polyhedra, combinatorial by Gallier J.

By Gallier J.

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By Gallier J.

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Extra info for Notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi diagrams and Delaunay triangulations

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2) As a subset of En cut out by a finite number of hyperplanes, more precisely, as the intersection of a finite number of (closed) half-spaces. As stated, these two definitions are not equivalent because (1) implies that a polyhedron is bounded, whereas (2) allows unbounded subsets. Now, if we require in (2) that the convex set A is bounded, it is quite clear for n = 2 that the two definitions (1) and (2) are equivalent; for n = 3, it is intuitively clear that definitions (1) and (2) are still equivalent, but proving this equivalence rigorously does not appear to be that easy.

An ), the equation of the polar hyperplane, a† , is a1 X1 + · · · + an Xn = 1. 38 CHAPTER 3. SEPARATION AND SUPPORTING HYPERPLANES Remark: As we noted, polarity in a Euclidean space suffers from the minor defect that the polar of the origin is undefined and, similarly, the pole of a hyperplane through the origin does not make sense. If we embed En into the projective space, Pn , by adding a “hyperplane at infinity” (a copy of Pn−1 ), thereby viewing Pn as the disjoint union Pn = En ∪ Pn−1 , then the polarity correspondence can be defined everywhere.

First, observe that (conv(Y ) + cone(V ))∗ = (conv(Y ∪ {Ω}) + cone(V ))∗ . If we pick Ω as an origin then we can represent the points in Y as vectors. The old origin is still denoted O and Ω is now denoted 0. The zero vector is denoted 0. If A = conv(Y ) + cone(V ) = {0}, let Y be the d × p matrix whose ith column is yi and let V is the d × q matrix whose j th column is vj . 13 says that (conv(Y ) + cone(V ))∗ = {x ∈ Rd | Y x ≤ 1, V x ≤ 0}. We write P (Y , 1; V , 0) = {x ∈ Rd | Y x ≤ 1, V x ≤ 0}.

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