### A closer look at lattice points in rational simplices.

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The notion of convex set for subsets of lattices in one particular case was introduced in [1], where it was used to study Paretto's principle in the theory of group choice. This notion is based on a betweenness relation due to Glivenko [2]. Betweenness is used very widely in lattice theory as basis for lattice geometry (see [3], and, especially [4 part 1]).In the present paper the relative notions of convexity are considered for subsets of an arbitrary lattice.In section 1 certain relative notions...

We first define the curvature indices of vertices of digital objects. Second, using these indices, we define the principal normal vectors of digital curves and surfaces. These definitions allow us to derive the Gauss-Bonnet theorem for digital objects. Third, we introduce curvature flow for isothetic polytopes defined in a digital space.

Let two lattices ${\Lambda}^{\text{'}},{\Lambda}^{\text{'}\text{'}}\subset {\mathbb{R}}^{s}$ have the same number of points on each hyperbolic surface $|x\u2081...{x}_{s}|=C$. We investigate the case when Λ’, Λ” are sublattices of ${\mathbb{Z}}^{s}$ of the same prime index and show that then Λ’ and Λ” must coincide up to renumbering the coordinate axes and changing their directions.

A Klein polyhedron is defined as the convex hull of nonzero lattice points inside an orthant of ${\mathbb{R}}^{n}$. It generalizes the concept of continued fraction. In this paper facets and edge stars of vertices of a Klein polyhedron are considered as multidimensional analogs of partial quotients and quantitative characteristics of these “partial quotients”, so called determinants, are defined. It is proved that the facets of all the ${2}^{n}$ Klein polyhedra generated by a lattice $\Lambda $ have uniformly bounded determinants...