A Steiner deltoid maintains constant area across all boundary points of an ellipse.
problem Finding curves associated with ellipses with constant area.
method Negative Pedal Curve of the Ellipse with respect to a boundary point M.
result The Steiner deltoid has constant area over all boundary points.
Proves Steiner and tube formulae for 3D contact sub-Riemannian surfaces.
problem Calculating surface properties in complex geometric structures.
method Develops a local Steiner formula for regular surfaces in 3D contact sub-Riemannian manifolds.
result Establishes a formula for surface expansion in arbitrary regions of contact sub-Riemannian manifolds.
New Lp-Steiner quermassintegrals defined from Steiner formula.
problem Defining new Lp-Steiner quermassintegrals. method Analogy to classical Steiner formula, investigating properties in convex bodies.
result Rotation and reflection invariant valuations in convex bodies.
New Steiner formula for Lp affine surface area in Minkowski theory.
problem Developing a new Steiner formula for Lp affine surface area. method Proving a new Steiner formula for the Lp affine surface area of a Minkowski outer parallel body. result New curvature measures with properties not previously seen in literature.
Finite-type k-surfaces in hyperbolic space have finite genus and cusp-like ends.
problem Characterizing the geometry of finite-type k-surfaces in hyperbolic space.
method Analyzing the asymptotic behavior and geometric properties of finite-type k-surfaces.
result Finite-type k-surfaces have well-defined Steiner geodesics and points, leading to new geometric identities.
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…
New proof of isoperimetric inequality using Steiner's formula.
problem Proving the isoperimetric inequality in the plane.
method Direct proof using Steiner's formula, bypassing domain existence.
result Establishes the isoperimetric inequality directly.
Improved bounds for isoperimetric deficit of convex sets.
problem Finding upper and lower bounds for the isoperimetric deficit of convex sets.
method Using the evolute, pedal curve, and Steiner point properties to derive inequalities.
result Strictly positive lower bounds for the isoperimetric deficit involving geometric properties of the convex set.
A 1-parameter family of Steiner chains has constant curvature moments.
problem Characterize the curvature moments of a 1-parameter family of Steiner chains.
method Proved constant curvature moments for k=3 using Descartes Circle Theorem; extended to spherical and hyperbolic geometries.
result First k-1 moments of curvatures remain constant in a 1-parameter family of Steiner chains.
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…
Infinite fractal tree solves shortest connection problem.
problem Finding the shortest connection for a fractal set.
method Constructing an infinite planar self-similar binary tree.
result The tree is the unique solution to the Steiner problem.
Symmetric TSP is structurally equivalent to a constrained Group Steiner Tree Problem.
problem Finding the shortest tour in a symmetric TSP.
method Structural equivalence between symmetric TSP and constrained Group Steiner Tree Problem.
result Maximizing net weight in the cGSTP is equivalent to minimizing the TSP tour length.
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 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 C1,1. 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…
The paper defines conic reach and shows polynomial parallel volume in the plane.
problem Understanding geometric properties of sets in the plane.
method Introducing conic reach and using local Steiner formula to show polynomial volume.
result Polynomial parallel volume of sets in the plane with conic reach.
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 …
The paper extends Busemann's inequalities to complex and quaternionic spaces.
problem Extending Busemann's inequalities to complex and quaternionic vector spaces.
method Proof leverages a monotonicity property under symmetrization with respect to complex or quaternionic hyperplanes.
result Standard Steiner symmetrization does not exhibit the monotonicity property in complex or quaternionic spaces.
Minimal surfaces created from tiny circle packings changes.
problem Creating minimal surfaces from circle packings.
method Parametrizing deformations of circle packings and relating them to discrete minimal surfaces.
result Every minimal surface of Koebe type can be extended to a general type minimal surface.
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.
Pedal curves derived from ellipses are invariant in area.
problem Finding invariant areas of pedal curves derived from ellipses.
method Analytical proof and explicit area expressions.
result Pedal curves derived from ellipses are invariant in area.
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…
Paper proposes an algorithm to reconstruct optimal model structure from graph adjacency matrix.
problem Optimal model structure reconstruction from weighted colored graph adjacency matrix.
method Uses prize-collecting Steiner tree algorithm to reconstruct minimum spanning tree.
result Demonstrates the effectiveness of the prize-collecting Steiner tree algorithm for model structure reconstruction.
NeuroSteiner uses neural networks to estimate wirelength more efficiently.
problem Minimizing wirelength in chip design.
method Neural model trained on synthesized nets to estimate WL.
result NeuroSteiner achieves 0.3% WL error at 60% faster than GeoSteiner.
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 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.
The Heisenberg group's curvature and Gauss-Bonnet theorem are explored using Riemannian approximation.
problem Defining curvature in the Heisenberg group for smooth surfaces and curves.
method Using a Riemannian approximation scheme to define sub-Riemannian Gaussian and signed geodesic curvatures.
result Proved a Heisenberg version of the Gauss-Bonnet theorem.
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…
We prove that the curvature flow of an embedded planar network of three curves connected through a triple junction, with fixed endpoints on the boundary of a given strictly convex domain, exists smooth until the lengths of the three curves stay far from zero. If this is the case for all times, then the evolution exists…
Solves Minkowski problem for affine invariant convex domains.
problem Finding convex sets with given area measures in affine spaces.
method Variational method using Steiner formula and covolume functional.
result Solves the affine invariant Minkowski problem.
The paper studies stability of discrete planar curves using variational methods.
problem Stability of discrete planar curves under area constraints.
method Unified interpretation of discrete curvatures, determination of equilibrium curves, stability analysis.
result Equilibrium curves for the length functional under area-constraint conditions are determined and their stability is studied.
Corrected an error in yield curve behavior models.
problem Error in the boundary expression for yield curve shapes.
method Revised the mathematical expression for yield curve behavior.
result Corrected the boundary expression for normal and humped yield curves.
Uniqueness of stable, non-smooth hypersurfaces with constant anisotropic mean curvature.
problem Identifying stable, non-smooth hypersurfaces with constant anisotropic mean curvature.
method Study of piecewise-smooth hypersurfaces with anisotropic energy, proving uniqueness of the Wulff shape under certain conditions.
result Closed stable equilibrium hypersurfaces are unique and the Wulff shape when the anisotropic energy density is twice continuously differentiable and convex.
There is a one-to-one correspondence between geometric lattices and the intersection lattices of arrangements of homotopy spheres. When the arrangements are essential and fully partitioned, Zaslavsky's enumeration of the cells of the arrangement still holds. An application of the theory shows that all minimal cellular …
Study finds the shortest triply periodic graph spanning a cubic lattice.
problem Finding the shortest periodic graph with a fixed volume.
method Analyzes the body centred cubic lattice and the gyroid surface.
result The shortest graph is the srs network with K4 quotient. Study on Lp affine surface areas and their inequalities for convex bodies.
problem Understanding weighted Lp affine surface areas in convex bodies. method Investigating valuations, isoperimetric inequalities, and connections to f divergences. result Established isoperimetric inequalities for weighted Lp affine surface areas. We let (M^m, g) be a closed smooth Riemannian manifold (m >1) with positive scalar curvature S_g, and prove that the Yamabe constant of (M \times R^n,g+g_E) is achieved by a metric in the conformal class of (g+g_E), where g_E is the Euclidean metric. We also show that the Yamabe quotient of (M \times R^n,g+g_E) is impr…
Higher chromatic numbers χs of simplicial complexes naturally generalize the chromatic number χ1 of a graph. In any fixed dimension d, the s-chromatic number χs of d-complexes can become arbitrarily large for s≤⌈d/2⌉ [6,18]. In contrast, χd+1=1, and only little is known on χs for …
Classical integral geometry takes place in Euclidean space, but one can attempt to imitate it in any other metric space. In particular, one can attempt this in R^n equipped with the metric derived from the p-norm. This has, in effect, been investigated intensively for 1<p<\infty, but not for p=1. We show that integral …
New metrics help predict Brownian motion on surfaces and higher dimensions.
problem Predicting Brownian motion on complex surfaces and higher dimensions.
method Developed new metrics (Uniform Drainage Metric) for surfaces and higher dimensions.
result Uniform Drainage Metric predicts Brownian motion's narrow escape time consistently.
New connection found between shape reconstruction methods and persistent homology.
problem Connecting shape reconstruction methods with persistent homology.
method Wrap complexes and lexicographic optimal homologous cycles.
result Lexicographically optimal homologous cycles are supported on Wrap complexes.
New set class preserves Fourier series terms for planar ovals, leading to isoperimetric inequalities.
problem Investigate geometric properties of kth Order Preserving Sets and ovals. method Introduce and analyze kth Order Preserving Sets and Midpoint Sets; study geometric properties and isoperimetric inequalities. result Established an isoperimetric-type inequality relating perimeter and area of ovals and their associated sets.
New curvature measures characterize non-convex Wulff shapes in normed spaces.
problem Characterizing non-convex sets with curvature measures.
method Extending curvature measures to non-convex and non-smooth sets in normed spaces.
result Finite unions of disjoint Wulff shapes are the only sets with proportional curvature measures.
The paper relates curvature loci of different manifold types through projections and normal sections.
problem Understanding the geometry of manifolds and their curvature loci.
method Using normal sections and projections to relate curvature loci of different manifold types.
result A commutative diagram of projections and normal sections that relates the curvature loci of different types of manifolds.
Paper studies weighted Fermat-Frechet problem for simplex edge lengths.
problem Finding optimal edge lengths for simplex deformations.
method Isometric embedding techniques for K-Space. result New variational method to solve weighted Fermat-Frechet problem.
New manifolds found without interior conjugate points.
problem Existence of interior conjugate points in hyperbolic manifolds.
method Construction of non-trapping asymptotically hyperbolic manifolds.
result Found manifolds without interior conjugate points.
Example shows not all conjugate points are bifurcation points in semi-Riemannian geodesics.
problem Determining which conjugate points in semi-Riemannian geodesics are bifurcation points.
method Revisiting and correcting an example by Musso, Pejsachowicz, and Portaluri.
result Every conjugate point on the improved example is a bifurcation point.
Proposes model-based approach for MI learning using point process theory.
problem Lack of statistical point pattern models in MI learning.
method Develops framework using point process theory for principled extensions of MI learning tasks.
result Tractable point pattern models and solutions for MI learning and decision making.
Estimator calculates surface curvature from point cloud samples.
problem Accurately estimating curvature from limited point cloud data.
method Algorithm using probability distribution and nearby points control.
result Controlled number of points ensures accurate curvature estimation.