arXiv research

Symmetric TSP is structurally equivalent to a constrained Group Steiner Tree Problem.

Infinite fractal tree solves shortest connection problem.

Paper proposes an algorithm to reconstruct optimal model structure from graph adjacency matrix.

NeuroSteiner uses neural networks to estimate wirelength more efficiently.

Proves Steiner and tube formulae for 3D contact sub-Riemannian surfaces.

New $L_p$-Steiner quermassintegrals defined from Steiner formula.

We give new characterisations of sets of positive reach and show that a closed hypersurface has positive reach if and only if it is of class $C^{1,1}$. These results are then used to prove new alternating Steiner formulæ for hypersurfaces of positive reach. Furthermore, it will turn out that every hypersurface that sat…

In this paper, a new approach of defining Steiner symmetrization of coercive convex functions is proposed and some fundamental properties of the new Steiner symmetrization are proved. Further, using the new Steiner symmetrization, we give a different approach to prove a functional version of the Blaschke-Santalo inequa…

We prove an analogue of the classical Steiner formula for the $L_p$ affine surface area of a Minkowski outer parallel body for any real parameters $p$. We show that the classical Steiner formula and the Steiner formula of Lutwak's dual Brunn Minkowski theory are special cases of this new Steiner formula. This new Stein…

A Steiner deltoid maintains constant area across all boundary points of an ellipse.

We consider the existence problem for `Steiner networks' (trivalent graphs with 120 degree angles at each junction) in strictly convex domains, with `Neumann' boundary conditions (orthogonal intersection with the domain boundary.) For each of the three possible combinatorial possibilities, sufficient conditions on the …

A Steiner type formula for continuous translation invariant Minkowski valuations is established. In combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations, this new Steiner type formula is used to obtain a family of Brunn-Minkowski type inequalities for rigid motion intertwi…

A Steiner chain of length k consists of k circles, tangent to two given non-intersecting circles (the parent circles) and tangent to each other in a cyclic pattern. The Steiner porism states that once a chain of k circles exists, there exists a 1-parameter family of such chains with the same parent circles that can be …

We give a new proof of the isoperimetric inequality in the plane, based on Steiner's formula for the area of a convex neighborhood. This proof establishes the isoperimetric inequality directly, without requiring that we separately establish the existence of an optimal domain. In doing so, this proof bypasses the main d…

We provide very general symmetrization theorems in arbitrary dimension and codimension, in products, warped products, and certain fiber bundles such as lens spaces, including Steiner, Schwarz, and spherical symmetrization and admitting density.

We prove the existence of self-similar expanding solutions of the curvature flow on planar networks where the initial configuration is any number of half-lines meeting at the origin. This generalizes recent work by Schnürer and Schulze which treats the case of three half-lines. There are multiple solutions, and these a…

The Gilbert-Steiner problem is a mass transportation problem, where the cost of the transportation depends on the network used to move the mass and it is proportional to a certain power of the "flow". In this paper, we introduce a new formulation of the problem, which turns it into the minimization of a convex function…

We use a Riemannnian approximation scheme to define a notion of $\textit{sub-Riemannian Gaussian curvature}$ for a Euclidean $C^{2}$-smooth surface in the Heisenberg group $\mathbb{H}$ away from characteristic points, and a notion of $\textit{sub-Riemannian signed geodesic curvature}$ for Euclidean $C^{2}$-smooth curve…

The paper extends Busemann's inequalities to complex and quaternionic spaces.

We establish a new symmetrization procedure for the isoperimetric problem in symmetric spaces of noncompact type. This symmetrization generalizes the well known Steiner symmetrization in euclidean space. In contrast to the classical construction the symmetrized domain is obtained by solving a nonlinear elliptic equatio…

Solves Minkowski problem for affine invariant convex domains.

In this paper, using functional Steiner symmetrizations, we show that Meyer and Pajor's proof of the Blaschke-Santalo inequality can be extended to the functional setting.

Paper solves isomorphism problem for specific Baumslag-Solitar groups.

Paper studies weighted Fermat-Frechet problem for simplex edge lengths.

We study a problem of geometric graph theory: We determine the triply periodic graph in Euclidean 3-space which minimizes length among all graphs spanning a fundamental domain of 3-space with the same volume. The minimizer is the so-called srs network with quotient the complete graph on four vertices $K_4$. The network…

For $0<k<1$, a finite-type $k$-surface in $3$-dimensional hyperbolic space is a complete, immersed surface of finite area and of constant extrinsic curvature equal to $k$. In [32], we showed that such surfaces have finite genus and finitely many cusp-like ends. Each of these cusps is asymptotic to an immersed cylinder …

We show that the discrete principal nets in quadrics of constant curvature that have constant mixed area mean curvature can be characterized by the existence of a Königs dual in a concentric quadric.

Let $Λ_0$ be an ordered abelian group. We show how an $\mathrm{ATF}(\mathbb{Z}\timesΛ_0)$ group -- that is, a group admitting a free affine action without inversions on a $\mathbb{Z}\timesΛ_0$-tree -- admits a natural graph of groups decomposition, where vertex groups inherit actions on $Λ_0$-trees. Using recent work o…

Bowditch's JSJ tree for splittings over 2-ended subgroups is a quasi-isometry invariant for 1-ended hyperbolic groups which are not cocompact Fuchsian. Our main result gives an explicit, computable "visual" construction of this tree for certain hyperbolic right-angled Coxeter groups. As an application of our constructi…

The paper studies stability of discrete planar curves using variational methods.

Given a simple closed plane curve $Γ$ of length $L$ enclosing a compact convex set $K$ of area $F$, Hurwitz found an upper bound for the isoperimetric deficit, namely $L^2-4πF\leq π|F_{e}|$, where $F_{e}$ is the algebraic area enclosed by the evolute of $Γ$. In this note we improve this inequality finding strictly posi…

Characterizes fundamental groups of disjointly tree-graded spaces.

We prove that an arbitrary right-angled Artin group $G$ admits a quasi-isometric group embedding into a right-angled Artin group defined by the opposite graph of a tree. Consequently, $G$ admits quasi-isometric group embeddings into a pure braid group and into the area-preserving diffeomorphism groups of the 2--disk an…

New groups defined that act on trees without repeating.

We study the problem of learning a latent tree graphical model where samples are available only from a subset of variables. We propose two consistent and computationally efficient algorithms for learning minimal latent trees, that is, trees without any redundant hidden nodes. Unlike many existing methods, the observed …

The paper studies acylindrical actions on trees and proves acylindrical hyperbolicity of Baumslag-Solitar groups.

Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asy…

Study on embedding tree products into groups, distinguishing them.

In this article we study the K- and L-theory of groups acting on trees. We consider the problem in the context of the fibered isomorphism conjecture of Farrell and Jones. We show that in the class of residually finite groups it is enough to prove the conjecture for finitely presented groups with one end. Also, we deduc…

Reduces conjecture to tree-based Artin groups.

Study of tree automorphisms via arc-stabilizers.

A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general construction of many actions of groups on quasi-trees. The groups we can handle include non-elementary (relatively) hyperbolic groups, rank 1 CAT(0) groups, mapping class groups and Out(Fn). As an application, we show that mapping clas…

We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an R-tree. We characterize lacunary hyperbolic groups as direct limits of Gromov hyperbolic groups satisfying certain restrictions on the hyperbolicity constants and injectivity radii. Using central extensions of lacunary hyperboli…

Conditions for reducing quasi-actions to tree actions and group properties.

We study a notion of deformation for simplicial trees with group actions (G-trees). Here G is a fixed, arbitrary group. Two G-trees are related by a deformation if there is a finite sequence of collapse and expansion moves joining them. We show that this relation on the set of G-trees has several characterizations, in …

Non-proper surface group action on product of trees found.

ODTLearn learns optimal decision trees for predictive and prescriptive tasks.

Study groups acting on trees with specific local actions, proving cohomology vanishing or infinite.