Mathematics

In the course of classifying generic sparse polynomial systems which are solvable in radicals, Esterov recently showed that the volume of the Minkowski sum P 1 +⋯+ P d of d -dimensional lattice polytopes is bounded from above by a function of order O( m 2 d ) , where m is the mixed volume of the tuple ( P 1 ,…, P d ) . This is a consequence of the well-known Aleksandrov-Fenchel inequality. Esterov also posed the problem of determining a sharper bound. We show how additional relations between mixed volumes can be employed to improve the bound to O( m d ) , which is asymptotically sharp. We furthermore prove a sharp exact upper bound in dimensions 2 and 3. Our results generalize to tuples of arbitrary convex bodies with volume at least one.

Several years ago the authors started looking at some problems of convex geometry from a more general point of view, replacing volume by an arbitrary measure. This approach led to new general properties of the Radon transform on convex bodies including an extension of the Busemann-Petty problem and a slicing inequality for arbitrary functions. The latter means that the sup-norm of the Radon transform of any probability density on a convex body of volume one is bounded from below by a positive constant depending only on the dimension. In this note, we prove an inequality that serves as an umbrella for these results

We study metric measure spaces that admit "thick" families of rectifiable curves or curve fragments, in the form of Alberti representations or curve families of positive modulus. We show that such spaces cannot be bi-Lipschitz embedded into any Euclidean space unless they admit some "infinitesimal splitting": their tangent spaces are bi-Lipschitz equivalent to product spaces of the form Z× R k for some k≥1 . We also provide applications to conformal dimension and give new proofs of some previously known non-embedding results.

In this article, we show that the Goldman-Iwahori metric on the space of all norms on a fixed vector space satisfies the Helly property for balls. On the non-Archimedean side, we deduce that most classical Bruhat-Tits buildings may be endowed with a natural piecewise ????metric which is injective. We also prove that most classical semisimple groups over non-Archimedean local fields act properly and cocompactly on Helly graphs. This gives another proof of biautomaticity for their uniform lattices. On the Archimedean side, we deduce that most classical symmetric spaces of non-compact type may be endowed with a natural piecewise ????metric which is coarsely Helly. We also prove that most classical semisimple groups over Archimedean local fields act properly and cocompactly on injective metric spaces. The only exception is the special linear group: if n?? and K is a local field, we show that SL(n,K) does not act properly and coboundedly on an injective metric space.

We investigate polytopes inscribed into a sphere that are normally equivalent to a given polytope P . We show that the associated space of polytopes, called the inscribed cone of P , is closed under Minkowski addition. Inscribed cones are interpreted as type cones of ideal hyperbolic polytopes and as deformation spaces of Delaunay subdivisions. In particular, testing if there is an inscribed polytope normally equivalent to P is polynomial time solvable. Normal equivalence is decided on the level of normal fans and we study the structure of inscribed cones for various classes of polytopes and fans, including simple, simplicial, and even. We classify (virtually) inscribable fans in dimension 2 as well as inscribable permutahedra and nestohedra. A second goal of the paper is to introduce inscribed virtual polytopes. Polytopes with a fixed normal fan N form a monoid with respect to Minkowski addition and the associated Grothendieck group is called the type space of N . Elements of the type space correspond to formal Minkowski differences and are naturally equipped with vertices and hence with a notion of inscribability. We show that inscribed virtual polytopes form a subgroup, which can be nontrivial even if N does not have actual inscribed polytopes. We relate inscribed virtual polytopes to routed particle trajectories, that is, piecewise linear trajectories of particles in a ball with restricted directions. The state spaces gives rise to connected groupoids generated by reflections, called reflection groupoids. The endomorphism groups of reflection groupoids can be thought of as discrete holonomy groups of the trajectories and we determine when they are reflection groups.

Nielsen's theorem states that any triangle can be inscribed in a planar Jordan curve. We prove a generalisation of this theorem, extending to any Jordan curve J embedded in R n , for a restricted set of triangles. We then conclude by investigating a condition under which a given point of J inscribes an equilateral triangle in particular.

This article surveys the θ -intermediate dimensions that were introduced recently which provide a parameterised continuum of dimensions that run from Hausdorff dimension when θ=0 to box-counting dimensions when θ=1 . We bring together diverse properties of intermediate dimensions which we illustrate by examples.

Consider the map S which sends a planar polygon P to a new polygon S(P) whose vertices are the intersection points of second nearest sides of P . This map is the inverse of the famous pentagram map. In this paper we investigate the dynamics of the map S . Namely, we address the question of whether a convex polygon stays convex under iterations of S . Computer experiments suggest that this almost never happens. We prove that indeed the set of polygons which remain convex under iterations of S has measure zero, and moreover it is an algebraic subvariety of codimension two. We also discuss the equations cutting out this subvariety, as well as their geometric meaning in the case of pentagons.

We describe invariants of centers of ellipse-inscribed triangle families with two vertices fixed to the ellipse boundary and a third one which sweeps it. We prove that: (i) if a triangle center is a fixed affine combination of barycenter and orthocenter, its locus is an ellipse; (ii) and that over the family of said affine combinations, the centers of said loci sweep a line; (iii) over the family of parallel fixed vertices, said loci rigidly translate along a second line. Additionally, we study invariants of the envelope of elliptic loci over combinations of two fixed vertices on the ellipse.

The point pair function p G defined in a domain G⊊ R n is shown to be a quasi-metric and its other properties are studied. For a convex domain G⊊ R n , a new intrinsic quasi-metric called the function w G is introduced. Several sharp results are established for these two quasi-metrics, and their connection to the triangular ratio metric is studied.