On the lattice rest of a convex body in ? s

Abstract Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$ . In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map $R(\cdot )$ is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has $C^{1}$ boundary. Furthermore, we prove that the map $R(\cdot )$ is surjective, extending a result of Kucherenko and Wolf.

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Bounds on the Lattice Point Enumerator via Slices and Projections

Discrete & Computational Geometry ◽

10.1007/s00454-021-00310-7 ◽

2021 ◽

Author(s):

Ansgar Freyer ◽

Martin Henk

Keyword(s):

Convex Body ◽

Lattice Point ◽

Geometric Mean ◽

Discrete Analogue ◽

Lattice Points ◽

General Context ◽

General Bound

AbstractGardner et al. posed the problem to find a discrete analogue of Meyer’s inequality bounding from below the volume of a convex body by the geometric mean of the volumes of its slices with the coordinate hyperplanes. Motivated by this problem, for which we provide a first general bound, we study in a more general context the question of bounding the number of lattice points of a convex body in terms of slices, as well as projections.

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An extremal property of the hypersphere

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100026542 ◽

1951 ◽

Vol 47 (1) ◽

pp. 245-247 ◽

Cited By ~ 33

Author(s):

A. M. Macbeath

Keyword(s):

Convex Body ◽

Extremal Property ◽

Plane Case ◽

Simple Application ◽

Plane Convex ◽

Plane Convex Body

It was shown by Sas (1) that, if K is a plane convex body, then it is possible to inscribe in K a convex n-gon occupying no less a fraction of its area than the regular n-gon occupies in its circumscribing circle. It is the object of this note to establish the n-dimensional analogue of Sas's result, giving incidentally an independent proof of the plane case. The proof is a simple application of the Steiner method of symmetrization.

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