xvi Notation Cones Cone(S) convex cone generated by S (1) σ rational convex polyhedral cone in NR (1) Span(σ) subspace spanned by σ (1) dim σ dimension of σ (1) σ∨ dual cone of σ (1) Relint(σ) relative interior of σ (1) Int(σ) interior of σ when Span(σ) = NR (1) σ⊥ set of m ∈ MR with m,σ = 0 (1) τ σ, τ ≺ σ τ is a face or proper face of σ (1) τ ∗ face of σ∨ dual to τ ⊆ σ, equals σ∨ ∩ τ ⊥ (1) Rays ρ 1-dimensional strongly convex cone (a ray) in NR (1) uρ minimal generator of ρ ∩ N, ρ a rational ray in NR (1) σ(1) rays of a strongly convex cone σ in NR (1) Lattices ZA lattice generated by A (1) Z A elements ∑s i=1 aimi ∈ ZA with ∑s i=1 ai = 0 (2) Nσ sublattice Z(σ ∩ N) = Span(σ) ∩ N (3) N(σ) quotient lattice N/Nσ (3) M(σ) dual lattice of N(σ), equals σ⊥ ∩ M (3) Fans Σ fan in NR (2,3) Σ(r) r-dimensional cones of Σ (3) Σmax maximal cones of Σ (3) Star(σ) star of σ, a fan in N(σ) (3) Σ∗(σ) star subdivision of Σ for σ ∈ Σ (3) Σ∗(v) star subdivision of Σ for v ∈ |Σ|∩ N primitive (11) Polytopes and Polyhedra Δn standard n-simplex in Rn (2) Conv(S) convex hull of S (1) dim P dimension of a polyhedron P (2) Q P, Q ≺ P Q is a face or proper face of P (2) P◦ dual or polar of a polytope (2) A + B Minkowski sum (2) kP multiple of a polytope or polyhedron (2)
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