Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that “open” polyhedra with triangular faces may not be unfoldable no matter how they are cut.

A well-studied problem is that of unfolding a convex polyhedron into a simple planar polygon. In this paper, we study the limits of unfoldability. We give an example of a polyhedron with convex faces that cannot be unfolded by cutting along its edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that "open" polyhedra with convex faces may not be unfoldable no matter how they are cut.

We develop the first nontrivial lower bounds on the complexity of online hyperplane and halfspace emptiness queries. Our lower bounds apply to a general class...

It is well known that for P and Q lattice polytopes, the Ehrhart polynomial of P ×Q satisfies LP ×Q(t) = LP (t)LQ(t). We show that there is a similar multiplicative relationship between the Ehrhart series for P, for Q, and for the free sum P ⊕ Q that holds when P is reflexive and Q contains 0 in its interior. Let P be a lattice polytope of dimension d, i.e. a convex polytope in Rn whose vertices are elements of Zn and whose affine span has dimension d. A remarkable theorem due to E. Ehrhart, [4], asserts that for non-negative integers t the number of lattice points in the tth dilate of P is given by a degree d polynomial in t denoted by LP (t) and called the Ehrhart polynomial of P. We let EhrP (x) = � LP (t)x t �d j=0 h∗j xj (1 − x) d+1 t≥0

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Abstract: A new way of constructing fusion bases (i.e., the set of inequalities governing fusion rules) out of fusion elementary couplings is presented. It relies on a polytope reinterpretation of the problem: the elementary couplings are associated to the vertices of the polytope while the inequalities defining the fusion basis are the facets. The symmetry group of the polytope associated to the lowest rank affine Lie algebras is found; it has order 24 for ̂su(2), 432 for ̂su(3) and quite surprisingly, it reduces to 36 for ̂su(4), while it is only of order 4 for ̂sp(4). This drastic reduction in the order of the symmetry group as the algebra gets more complicated is rooted in the presence of many linear relations between the elementary couplings that break most of the potential symmetries. For ̂su(2) and ̂su(3), it is shown that the fusion-basis defining inequalities can be generated from few (1 and 2 respectively) elementary ones. For ̂su(3), new symmetries of the fusion coefficients are found.

Abstract. A real representation of a finite group naturally determines a polytope, generalizing the well-known Birkhoff polytope. This paper determines the structure of the polytope corresponding to the natural permutation representation of a general Frobenius group. 1.

DMS-0401047 from the National Science Foundation. Abstract. The fourteen lectures in this book were prepared for the advanced undergraduate course at the Park City Mathematics Institute on Geometric Combinatorics in July 2004. They begin with the basics of polytope theory with an emphasis on geometry via the theory of Schlegel and Gale diagrams. The lectures lead up to secondary and state polytopes arising from point configurations. These polytopes are relatively recent constructs that have connections to several parts of mathematics such as combinatorics, commutative algebra, algebraic geometry and symplectic geometry. The treatment here constructs them from scratch and focuses on their geometric combinatorics. This book is meant to be accessible to undergraduates with a background in linear algebra.

Because of the hairy ball theorem, the only closed 2-manifold that supports a lattice