Volume 16, Issue 4

In this paper we study smooth metric measure spaces with boundary via the Bakry–Émery curvature and the weighted mean curvature of the boundary. We establish the weighted Laplacian comparison theorems and the upper bound estimates of the distance from any point of the manifold to its boundary. As applications, we derive lower bound estimates for the first Dirichlet eigenvalue.

Hilbert schemes of suitable smooth, projective threefold scrolls over the Hirzebruch surface 𝔽 e , e ≥ 2, are studied. An irreducible component of the Hilbert scheme parametrizing such varieties is shown to be generically smooth of the expected dimension, and the general point of such a component is described.

Given a point S (the light position) in ℙ 3 and an algebraic surface 𝓩 (the mirror) of ℙ 3 , the caustic by reflection Σ S (𝓩) of 𝓩 from S is the Zariski closure of the envelope of the reflected lines 𝓡 m obtained by reflection of ( Sm ) on 𝓩 at m ∈ 𝓩. We use the ramification method to identify Σ S (𝓩) with the Zariski closure of the image, by a rational map, of an algebraic 2-covering space of 𝓩. We also give a general formula for the degree (with multiplicity) of caustics (by reflection) of algebraic surfaces of ℙ 3 .

In 2008 R. Connelly asked how one should place n small disks of radius r to cover the largest possible area of a disk of radius R > r . More specifically, is there always an optimal configuration with n-fold rotational symmetry for small values of n ? The answer is known to be positive for n = 2, negative for n = 5, and it has been conjectured to be positive for n = 3 and 4. In this paper, we present a systematic way to list all possible combinatorial structures of optimal configurations, and we prove that for n = 3 there is always an optimal configuration with rotational symmetry of order three.

Our first result says that every real or complex infinite-dimensional normed space has an unbounded absolutely convex and absorbing subset with empty interior. As a consequence, a real normed space is finite-dimensional if and only if every convex subset containing 0 whose linear span is the whole space has non-empty interior. In our second result we prove that every real or complex separable normed space with dimension greater than 1 contains a balanced and absorbing subset with empty interior which is dense in the unit ball. Explicit constructions of these subsets are given.

The action of a Singer cyclic group of order q 2 + 1 on the Hermitian surface H(3,q2)$\mathcal{H}(3, q^2)$, q even, is investigated. Infinite families of hyperovals of size 2( q 2 + 1), q even, are then constructed.

We establish a fixed point theoremfor a certain type of non-expandingmaps in Euclidean buildings, which is inspired by a theorem of Laffaille in p -adic Hodge theory [10, Theorem 3.2].

A concept of nondegenerate basic closed semi-algebraic sets in ℝ n is introduced. These are unbounded closed semi-algebraic sets for which we obtain some representations of polynomials with positive infima (the polynomials are further assumed to be bounded if n >2) and solutions of the moment problem. The key to obtain these results is an explicit description of the algebra of bounded polynomials on a nondegenerate basic semi-algebraic set via the combinatorial information of the Newton polyhedron corresponding to the generators of the semi-algebraic set.

The study of the graph diameter of polytopes is a classical open problem in polyhedral geometry and the theory of linear optimization. In this paper we continue the investigation initiated in [5] by introducing a vast hierarchy of generalizations to the notion of graph diameter. This hierarchy provides some interesting lower bounds for the usual graph diameter. After explaining the structure of the hierarchy and discussing these bounds, we focus on clearly explaining the differences and similarities among the many diameter notions of our hierarchy. Finally, we fully characterize the hierarchy in dimension two. It collapses into fewer categories, for which we exhibit the ranges of values that can be realized as diameters.