We study semilattices whose diagrams are trees. First, we characterize them as semilattices whose convex subsemilattices form a convex geometry, or, equivalently, the closure induced by convex subsemilattices is antiexchange. Then we give lattice theoretic and two graph theoretic characterizations of atomistic semilattices with tree diagrams.

Weakly convex polyhedra which are star-shaped with respect to one of their vertices are infinitesimally rigid. This is a partial answer to the question whether every decomposable weakly convex polyhedron is infinitesimally rigid. The proof uses a recent result of Izmestiev on the geometry of convex caps.

We consider the problem of online convex optimization against an arbitrary adversary with bandit feedback, known as bandit convex optimization. We give the first Õ( √ T )-regret algorithm for this setting based on a novel application of the ellipsoid method to online learning. This bound is known to be tight up to logarithmic factors. Our analysis introduces new tools in discrete convex geometry.

A main aim of this paper is to make connections between two well – but up to now independently – developed theories, the theory of choice functions and the theory of closure operators. It is shown that the classes of ordinally rationalizable and path independent choice functions are related to the classes of distributive and anti-exchange closures.  1999 Elsevier Science B.V. All rights reserved.

STEINER POINTS FOR CONVEX POLYTOPES CHRISTIAN BERG Let & denote the set of all convex polytopes, degenerate or not, in ^-dimensional Euclidean space E. An abstract Steiner point for convex polytopes in E is a mapping S:2P-+E satisfying S(P+Q) = S(P) + S(Q) for all P, Qe0*, (1) addition on the left being Minkowski addition of convex sets, and S{a(P)) = o{S(P)) (2) for all Pe&* and all similarity...

A new algorithm for the determination of the relative convex hull in the plane of a simple polygon A with respect to another simple polygon B which contains A, is proposed. The relative convex hull is also known as geodesic convex hull, and the problem of its determination in the plane is equivalent to find the shortest curve among all Jordan curves lying in the difference set of B and A and en...

We consider a planar convex body C and we prove several analogs of Roth's theorem on irregularities distribution. When ∂ is 2 regardless curvature, that for every set P N points in T have the sharp bound ∫ 0 1 | card ( ∩ τ + t ) − d ⩾ c / . only piecewise not polygon 5 also give whole range intermediate results between Our proofs depend lemma Cassels-Montgomery, ad hoc constructions finite poin...

Recently, a unique class of local Higher Spin Gravities with propagating massless fields in 4d – Chiral Gravity was given covariant formulation both flat and (A)dS4 spacetimes at the level equations motion. We unfold corresponding homological perturbation theory as to explicitly obtain all interaction vertices. The vertices reveal remarkable simplicity after an appropriate change variables. Sim...