Generalized flatness constants, spanning lattice polytopes, and the Gromov width

In this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently large width contain a unimodular copy of a standard simplex. Following an argument of Eisenbrand and Shmonin, we prove that every lattice polytope contains a minimal generating set of the affine lattice spanned by its lattice points such that the number of generators (and the lattice width of their convex hull) is bounded by a constant which only depends on the dimension. We also discuss relations to recent results on spanning lattice polytopes and how our results could be viewed as the beginning of the study of generalized flatness constants. Regarding symplectic geometry, we point out how the lattice width of a Delzant polytope is related to upper and lower bounds on the Gromov width of its associated symplectic toric manifold. Throughout, we include several open questions.

Visualization of a Standing Lattice of Scattered Light on Spherical Water Droplets under an Acoustic Field Using Biogenic Microparticles

Micromanipulation using acoustic sound is a promising technique for drug delivery, cell manipulation, biosensors, and microfluidic devices. Additionally, the visualization of acoustic fields by advanced optical measurement techniques can be combined with this micromanipulation technique. The present study reveals that a lattice pattern of reflected light appears on the surface of water droplets containing microparticles when the droplets are exposed to audible sound in the range of 1900 to 10000 Hz. A piezoelectric membrane providing an audible acoustic field induced a stream of microparticles on which the lattice pattern overlapped, with the appearance of a standing wave. The effects of microparticles, including BaSO4, TiO2, and guanine platelets derived from fish scales, on the formation of the lattice pattern were investigated. These three types of microparticles in water enabled a visualization of the vortex streams and generated a lattice pattern of reflected light. The guanine platelets exhibited the most precise lattice pattern over the droplet surface, with a lattice width of 100 to 200 μm. This phenomenon may provide a new tool for detecting and manipulating micro vortex flows in the aqueous chamber of a microfluidic device combined with an acoustic transducer.

Complexity of linear relaxations in integer programming

For a setXof integer points in a polyhedron, the smallest number of facets of any polyhedron whose set of integer points coincides with Xis called the relaxation complexity $${{\,\mathrm{rc}\,}}(X)$$rc(X). This parameter, introduced by Kaibel & Weltge (2015), captures the complexity of linear descriptions of Xwithout using auxiliary variables. Using tools from combinatorics, geometry of numbers, and quantifier elimination, we make progress on several open questions regarding$${{\,\mathrm{rc}\,}}(X)$$rc(X)and its variant$${{\,\mathrm{rc}\,}}_\mathbb {Q}(X)$$rcQ(X), restricting the descriptions of Xto rational polyhedra. As our main results we show that$${{\,\mathrm{rc}\,}}(X) = {{\,\mathrm{rc}\,}}_\mathbb {Q}(X)$$rc(X)=rcQ(X)when: (a)Xis at most four-dimensional, (b)Xrepresents every residue class in$$(\mathbb {Z}/2\mathbb {Z})^d$$(Z/2Z)d, (c) the convex hull of Xcontains an interior integer point, or (d) the lattice-width of Xis above a certain threshold. Additionally,$${{\,\mathrm{rc}\,}}(X)$$rc(X)can be algorithmically computed when Xis at most three-dimensional, orXsatisfies one of the conditions (b), (c), or (d) above. Moreover, we obtain an improved lower bound on$${{\,\mathrm{rc}\,}}(X)$$rc(X)in terms of the dimension of X.