Steinitz's Theorem Research Articles

Quantitative Steinitz theorem: A polynomial bound

AbstractThe classical Steinitz theorem states that if the origin belongs to the interior of the convex hull of a set , then there are at most points of whose convex hull contains the origin in the interior. Bárány, Katchalski, and Pach proved the following quantitative version of Steinitz's theorem. Let be a convex polytope in containing the standard Euclidean unit ball . Then there exist at most vertices of whose convex hull satisfies with . They conjectured that holds with a universal constant . We prove , the first polynomial lower bound on . Furthermore, we show that is not greater than .

An analogue of a theorem of Steinitz for ball polyhedra in mathbb {R}^3

Steinitz’s theorem states that a graph G is the edge-graph of a 3-dimensional convex polyhedron if and only if, G is simple, plane and 3-connected. We prove an analogue of this theorem for ball polyhedra, that is, for intersections of finitely many unit balls in mathbb {R}^3.

Polycube Shape Space

AbstractThere are many methods proposed for generating polycube polyhedrons, but it lacks the study about the possibility of generating polycube polyhedrons. In this paper, we prove a theorem for characterizing the necessary condition for the skeleton graph of a polycube polyhedron, by which Steinitz's theorem for convex polyhedra and Eppstein's theorem for simple orthogonal polyhedra are generalized to polycube polyhedra of any genus and with non‐simply connected faces. Based on our theorem, we present a faster linear algorithm to determine the dimensions of the polycube shape space for a valid graph, for all its possible polycube polyhedrons. We also propose a quadratic optimization method to generate embedding polycube polyhedrons with interactive assistance. Finally, we provide a graph‐based framework for polycube mesh generation, quadrangulation, and all‐hex meshing to demonstrate the utility and applicability of our approach.

Levy–Steinitz theorem and achievement sets of conditionally convergent series on the real plane

Levy–Steinitz theorem characterizes the sum range of conditionally convergent series, that is a set of all its convergent rearrangements; in finitely dimensional spaces – it is an affine subspace. An achievement of a series is a set of all its subsums. We study the properties of achievement sets of series whose sum range is the whole plane. It turns out that it varies on the number of Levy vectors of a series.

Extendability to summable ideals

We continue our work on the ideal version of the Levy–Steinitz theorem on conditionally convergent series of vectors. In particular, we prove that for each series \({\sum_{n\in\omega}v_n}\), \({(v_n)_{n\in\omega} \subset\mathbb{R}^2}\), such that its sum range is \({\mathbb{R}^2}\) and its set of Levy vectors is of power at least 3, it is possible to find \({A\in\mathcal{I}}\) such that the sum range of \({\sum_{n\in A}v_n}\) is still \({\mathbb{R}^2}\), for some proper ideal \({\mathcal{I}\subset\mathcal{P}(\omega)}\).

Rearranging series of vectors on a small set

We consider a variation of the Lévy–Steinitz theorem where one rearranges only a small number of terms. In particular, we study the set of obtainable rearrangements of a multidimensional series by a permutation σ:ω→ω such that {n∈ω:σ(n)≠n}∈I for some proper ideal I⊂P(ω).

Model Theoretical Generalization of Steinitz’s TheoremDOI: 10.5007/1808-1711.2011v15n1p107

Infinitary languages are used to prove that any strong isomorphism of substruc- tures of isomorphic structures can be extended to an isomorphism of the structures. If the structures are models of a theory that has quantifier elimination, any isomorphism of sub- structures is strong. This theorem is a partial generalization of Steinitz's theorem for alge- braically closed fields and has as special case the analogous theorem for differentially closed fields. In this note, we announce results which will be proved elsewhere.

How to Exhibit Toroidal Maps in Space

Steinitz's theorem states that a graph is the 1-skeleton of a convex polyhedron if and only if it is 3-connected and planar. The polyhedron is called a geometric realization of the embedded graph. Its faces are bounded by convex polygons whose points are coplanar. A map on the torus does not necessarily have such a geometric realization. In this paper we relax the condition that faces are the convex hull of coplanar points. We require instead that the convex hull of the points on a face can be projected onto a plane so that the boundary of the convex hull of the projected points is the image of the boundary of the face. We also require that the interiors of the convex hulls of different faces do not intersect. Call this an exhibition of the map. A map is polyhedral if the intersection of any two closed faces is simply connected. Our main result is that every polyhedral toroidal map can be exhibited. As a corollary, every toroidal triangulation has a geometric realization.

On a conjecture related to geometric routing

We conjecture that any planar 3-connected graph can be embedded in the plane in such a way that for any nodes s and t, there is a path from s to t such that the Euclidean distance to t decreases monotonically along the path. A consequence of this conjecture would be that in any ad hoc network containing such a graph as a spanning subgraph, two-dimensional virtual coordinates for the nodes can be found for which the method of purely greedy geographic routing is guaranteed to work. We discuss this conjecture and its equivalent forms show that its hypothesis is as weak as possible, and show a result delimiting the applicability of our approach: any 3-connected K 3 , 3 -free graph has a planar 3-connected spanning subgraph. We also present two alternative versions of greedy routing on virtual coordinates that provably work. Using Steinitz's theorem we show that any 3-connected planar graph can be embedded in three dimensions so that greedy routing works, albeit with a modified notion of distance; we present experimental evidence that this scheme can be implemented effectively in practice. We also present a simple but provably robust version of greedy routing that works for any graph with a 3-connected planar spanning subgraph.

On the quantitative steinitz theorem in the plane

We prove that for anyk ź 4, any setX of points in the plane, and any pointP ź interior conv(X), there is a subsetY (X of at mostk points such that if conv(X) contains a disk with radiusr aroundP, then conv(Y) contains a disk with radius [cos(2/(k + 1))ź]/[cos(1/(k + 1))ź]r aroundP. This generalizes the quantitative Steinitz theorem in the plane and proves a conjecture of Barany and Heppes.

On the exact constant in the quantitative Steinitz theorem in the plane

We determine the maximal value ofr with the following property. If the convex hull of a set inR 2 contains a unit circleB, then a subset of at most four points can be selected so that the convex hull of this subset contains the circle of radiusr concentric withB. That the result is sharp is shown by the example when the original set is the set of vertices of a regular pentagon circumscribed aroundB.

Quantitative Steinitz's theorems with applications to multifingered grasping

We prove the following quantitative form of a classical theorem of Steintiz: Letm be sufficiently large. If the convex hull of a subsetS of Euclideand-space contains a unit ball centered on the origin, then there is a subset ofS with at mostm points whose convex hull contains a solid ball also centered on the origin and havingresidual radius $$1 - 3d\left( {\frac{{2d^2 }}{m}} \right)^{2/(d - 1)} .$$ The casem=2d was first considered by Baranyet al. [1]. We also show an upper bound on the achievable radius: the residual radius must be less than $$1 - \frac{1}{{17}}\left( {\frac{{2d^2 }}{m}} \right)^{2/(d - 1)} .$$ These results have applications in the problem of computing the so-calledclosure grasps by anm-fingered robot hand. The above quantitative form of Steinitz's theorem gives a notion ofefficiency for closure grasps. The theorem also gives rise to some new problems in computational geometry. We present some efficient algorithms for these problems, especially in the two-dimensional case.

Direct sums of ideals

We survey various existence and uniqueness theorems for decompositions of finitely generated modules over commutative rings, as direct sums of ideals of the ring. These theorems generalize a theorem of Steinitz published in 1912. Much of the paper is expository. The main new result is the following uniqueness theorem (well known to be true for integral domains): Let A i , B i be ideals of the commutative ring R, and suppose that the R-modules A 1 ⊕⋯⊕ A m and B 1 ⊕⋯⊕ B m are isomorphic. Then the ideal products A 1⋯ A m and B 1⋯ B m are isomorphic as R-modules.

The Steinitz theorem on rearrangement of series for nuclear spaces.

The Steinitz theorem on rearrangement of series for nuclear spaces.

5-connected 3-polytopes are refinements of octahedra

It is a consequence of a theorem of Steinitz that the boundary of every convex 3-dimensional polytope is a refinement of the boundary of the tetrahedron. We show that if the graph of the polytope is 5-connected then its boundary is a refinement of the boundary of an octahedron.

The asymptotic number of convex polyhedra

We obtain an asymptotic formula for the number of combinatorially distinct convex polyhedra with n edges. 1. Introduction. The number of combinatorial polyhedra has been studied by many authors, perhaps first of all by Euler. A very interesting account up to 1975 has been given by P. J. Frederico (2). By a well-known theorem of Steinitz a convex polyhedron is combinatorially equivalent to a 3-connected planar map (with more than three edges). B. Griinbaum's proof (3) is the most elegant known to these authors. Indeed Mani (5) has shown that for each 3-connected planar map M there is a convex polyhedron whose 1-skeleton is isomorphic to M and such that every automorphism of M is induced by an isometry of the convex polyhedron. The enumeration of planar maps has progressed considerably following the breakthrough of W. T. Tutte in the early sixties. Tutte (6) has given a survey of the methods and results known up to 1973. Until quite recently most of the progress concerned the enumeration of rooted planar maps. A planar map is said to be rooted when one directed edge in it is distinguished as the root and when two sides of the root are distinguished as left and right. In fact Tutte (7) has shown that the number R(n) of rooted 3-connected planar maps with n edges is asymptotic to

On a Theorem of Steinitz and Levy

Let $\sum \nolimits _{n \in \omega } {h(n)}$ be a conditionally convergent series in a real Banach space B. Let $S(h)$ denote the set of sums of the convergent rearrangements of this series. A well-known theorem of Riemann states that $S(h) = B$ if $B = R$, the reals. A generalization of Riemann’s Theorem, due independently to Levy [L] and Steinitz [S], states that if B is finite dimensional, then $S(h)$ is a linear manifold in B of dimension $> 0$. Another generalization of Riemann’s Theorem [M] can be stated as an instance of the Levy-Steinitz Theorem in the Banach space of regulated real functions on the unit interval I. This instance generalizes to the Banach space of regulated B-valued functions on I, where B is finite dimensional, implying a generalization of the Levy-Steinitz Theorem.

On a theorem of Steinitz and Levy

Let ∑ n ∈ ω h ( n ) \sum \nolimits _{n\,\, \in \,\omega } {h(n)} be a conditionally convergent series in a real Banach space B. Let S ( h ) S(h) denote the set of sums of the convergent rearrangements of this series. A well-known theorem of Riemann states that S ( h ) = B S(h)\, = \,B if B = R B\, = \,R , the reals. A generalization of Riemann’s Theorem, due independently to Levy [L] and Steinitz [S], states that if B is finite dimensional, then S ( h ) S(h) is a linear manifold in B of dimension > 0 > \,0 . Another generalization of Riemann’s Theorem [M] can be stated as an instance of the Levy-Steinitz Theorem in the Banach space of regulated real functions on the unit interval I. This instance generalizes to the Banach space of regulated B-valued functions on I, where B is finite dimensional, implying a generalization of the Levy-Steinitz Theorem.

Conditionally convergent series in a uniformly smooth Banach space

The theorem of Steinitz on the form of the set of points which are sums of convergent rearrangements of a given series is extended to series Σxk in the uniformly smooth Banach space X with modulus of smoothness ρ(t), satisfying the condition Σρ(¦|xk¦|) < ∞.

Decomposing pairs of modules

(2) M= Ml GD*.*. Ml with f(M )U=i. Further, we can ask for which decompositions M-M1 = M D Mn can we choose Ml Mi in (2)? We will call an R-module U semiprimary if the ring of multiplications of R on U/rad U is semisimple with minimum condition and U is finitely generated. (All modules considered in this paper will be unitary.) Our Main Theorem 1.2 states that if f:M-M1 C) ...M, onto U = U1 + * + U,, is a homomorphism of R-modules with M projective and each Ui semiprimary, and if, for each i there is some homomorphism of Mi onto Ui, then there is a decomposition (2) for which M M. Note that the sum U1 + *. need not be direct. The main theorem is applied to sharpen a theorem of Steinitz, Chevalley, and others [7, Theorem 22.12] which states that if N is a submodule of a finitely generated, torsion-frr e (hence projective) module M over a Dedekind domain R such that A '/N is a torsion module, then there exist decompositions M = Mi an .. D Mn (Maan ideal of R) and N-EIM, e .. 3 E,Mn (Ei = an ideal). The present contribution to this theorem (Corollary1.10) is that M1, -., Mn -1 can be chosen isomorphic to arbitrary nonzero ideals and the ideals Ei can be chosen subject only to the restriction MIN RIE1 E0 o D R/E, This sharpened version of the theorem turns out to have a more abstract (and slightly more general) formulation (Corollary 1.9): if Ni is a submodule of a projective module Mi (still over a Dedekind domain) such that M1/N1 M2/N2= finitely generated, then M1 M2 if and only if N1 N2; and when the conditions