When the intersection ∩iHi is bounded, it forms a convex polytope P ; otherwise, it is a polyhedron.
Spaces of polytopes and cobordism of quasitoric manifolds
This last polyhedron is convex and, by construction, it has a boundary limited by hyperplanes which are rational with respect to X .
Vector partition function and generalized Dahmen-Micchelli spaces
To start with let us recall the algebraic-geometric construction of a toric variety from a convex integral polyhedron σ ⊆ MR , the intersection of ﬁnitely many closed halfspaces.
An invitation to toric degenerations
Now given a convex integral polyhedron σ ⊆ MR , with dim σ = n for simplicity, and a face τ ⊆ σ we obtain the aﬃne toric variety Uτ := Spec (cid:0)C[Kτ σ ∩ M ](cid:1).
An invitation to toric degenerations
Thus ˜σ is equivalent to a polyhedral decomposition P of the convex integral polyhedron σ together with a function ϕ on σ that is piecewise aﬃne and strictly convex with respect to P and takes integral values at the vertices of P .
An invitation to toric degenerations
Here, P (F ) denotes the distributional perimeter of F , a quantity that agrees with Hn−1(∂F ) whenever F is either an open set with C 1 -boundary, a convex set, or a polyhedron.
Quantitative stability in the isodiametric inequality via the isoperimetric inequality
The map to be coloured may well be taken as drawn on the surface of a ball (as in a globe) rather than on a sheet of paper. A little thought shows that, without loss of generality, the countries may be taken to be the faces of a convex polyhedron inscribed in the ball.
Higher dimensional analogues of the map colouring problem
It is not hard to see that for any ﬁxed set of vik , (3.2) deﬁnes a convex polyhedron in x-space that contains in its boundary all x ∈ Rq such that xk ∈ {0, 1}, k ∈ Q, hence is suitable for generating intersection cuts.
Intersection cuts from multiple rows: a disjunctive programming approach
Conversely, any abstract polyhedron P ∗ with weighted edges satisfying the conditions 1-3 is the Poincar´e dual of a convex ideal polyhedron P with the exterior dihedral angles equal to the weights.
Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli space
Each 1-dimensional cone ρ ∈ Σ corresponds to a Weil divisor Dρ ⊂ X . A divisor D = Pρ aρDρ gives a (possibly unbounded) convex polyhedron ∆D = {m ∈ MR : hm, ρi ≥ −aρ} ⊂ MR .
Recent developments in toric geometry
In this case, ∆D is a n-dimensional integral convex polytope (= bounded polyhedron) which is combinatorially dual to Σ, i.e., facets of ∆D (faces of dimension n − 1) correspond to 1-dimensional cones ρ ∈ Σ and, more generally, i-dimensional faces of ∆D correspond to (n − i)-dimensional cones of Σ.
Recent developments in toric geometry
Let P be a convex polyhedron in the spherical space S n , Euclidean space IE n or hyperbolic space IH n .
Spherical simplices generating discrete reflection groups
Let E ⊂ [0, 1]d . i) Suppose that K is a symmetric convex polyhedron and that the Hausdorff dimension of E is greater than 1.
K-distance sets, Falconer conjecture, and discrete analogs
The boundary of D in Ω(G) is a piecewise-smooth (polyhedron) submanifold in Ω(G), divided into a union of smooth submanifolds (convex polygons) which are cal led faces.
A wild knot $\mathbb{S}^{2}\hookrightarrow\mathbb{S}^{4}$ as limit set of a Kleinian Group: Indra's pearls in four dimensions
The proof of this can be obtained from the fact that if we are given a closed convex polyhedron then every point in the relative interior can be written as a convex combination of the extreme points with the coeﬃcients > 0 which are inﬁnitely differentiable functions of the point.
A priori estimates of smoothness of solutions to difference Bellman equations with linear and quasilinear operators
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