On densities of lattice arrangements intersecting every i-dimensional affine subspace

In 1978, Makai Jr. established a remarkable connection between the volume-product of a convex body, its maximal lattice packing density and the minimal density of a lattice arrangement of its polar body intersecting every affine hyperplane. Consequently, he formulated a conjecture that can be seen as a dual analog of Minkowski's fundamental theorem, and which is strongly linked to the well-known Mahler-conjecture. Based on the covering minima of Kannan & Lov\'asz and a problem posed by Fejes T\'oth, we arrange Makai Jr.'s conjecture into a wider context and investigate densities of lattice arrangements of convex bodies intersecting every i-dimensional affine subspace. Then it becomes natural also to formulate and study a dual analog to Minkowski's second fundamental theorem. As our main results, we derive meaningful asymptotic lower bounds for the densities of such arrangements, and furthermore, we solve the problems exactly for the special, yet important, class of unconditional convex bodies.Comment: 19 page

A remark on perimeter-diameter and perimeter-circumradius inequalities under lattice constraints

In this note, we study several inequalities involving geometric functionals for lattice point-free planar convex sets. We focus on the previously not addressed cases perimeter--diameter and perimeter--circumradius

Mechanisms of Action and Targets of Nitric Oxide in the Oculomotor System

Nitric oxide (NO) production by neurons in the prepositus hypoglossi (PH) nucleus is necessary for the normal performance of eye movements in alert animals. In this study, the mechanism(s) of action of NO in the oculomotor system has been investigated. Spontaneous and vestibularly induced eye movements were recorded in alert cats before and after microinjections in the PH nucleus of drugs affecting the NO–cGMP pathway. The cellular sources and targets of NO were also studied by immunohistochemical detection of neuronal NO synthase (NOS) and NO-sensitive guanylyl cyclase, respectively. Injections of NOS inhibitors produced alterations of eye velocity, but not of eye position, for both spontaneous and vestibularly induced eye movements, suggesting that NO produced by PH neurons is involved in the processing of velocity signals but not in the eye position generation. The effect of neuronal NO is probably exerted on a rich cGMP-producing neuropil dorsal to the nitrergic somas in the PH nucleus. On the other hand, local injections of NO donors or 8-Br-cGMP produced alterations of eye velocity during both spontaneous eye movements and vestibulo-ocular reflex (VOR), as well as changes in eye position generation exclusively during spontaneous eye movements. The target of this additional effect of exogenous NO is probably a well defined group of NO-sensitive cGMP-producing neurons located between the PH and the medial vestibular nuclei. These cells could be involved in the generation of eye position signals during spontaneous eye movements but not during the VOR.Fondo de Investigación Sanitaria Grants 94/0388 and 97/2054Comunidad Autónoma de Madrid Grant 08.5/0019/1997Dirección General de Investigación Científica y Technológica Grant PB 93–117