The paper classifies compact hyperbolic Coxeter polytopes and improves upper bounds.
problem Classifying compact hyperbolic Coxeter polytopes and understanding their combinatorial properties.
method Study of imes0-products of Lannér diagrams, proving superhyperbolic properties, and analyzing Lannér subdiagrams. result Improved upper bounds on the dimension of compact hyperbolic Coxeter polytopes.
In this paper we study a new combinatorial invariant of simple polytopes, which comes from toric topology. With each simple n-polytope P with m facets we can associate a moment-angle complex Z_P with a canonical action of the torus T^m. Then s(P) is the maximal dimension of a toric subgroup that acts freely on Z_P. The…
Generalizes crystallographic properties to all dimensions.
problem Analytic eigenfunctions in crystallographic groups.
method Algebraic, geometric, and analytic proofs.
result Equivalent conditions for real analytic eigenfunctions in crystallographic polytopes.
Article provides polytopes as dual unit balls of Thurston norms on 3-manifolds.
problem Understanding the dual unit ball shape of Thurston norms.
method Introduced a family of polytopes in Z^2g that can be dual unit balls of Thurston norms on 3-manifolds.
result Polytopes with mod 2 congruent vertices can be realized as dual unit balls of Thurston norms.
Unified approach to verify NN properties using ReLU's unique polytope structure.
problem Lack of robustness and interpretability in ReLU NNs for risk-sensitive applications.
method Identifying and traversing the local polytopes of ReLU NNs, developing an algorithm to verify properties.
result Unified approach to examine network behavior in risk-sensitive settings.
Proof of Gromov's theorem on convex polytopes with acute angles.
problem Gromov's conjecture on extremal scalar curvature of convex polytopes.
method Smoothing construction using Dirac operator techniques.
result Detailed proof of Gromov's theorem.
In [7], a notion of constant scalar curvature metrics on piecewise flat manifolds is defined. Such metrics are candidates for canonical metrics on discrete manifolds. In this paper, we define a class of vertex transitive metrics on certain triangulations of S3; namely, the boundary complexes of cyclic polyt…
In a series of papers the authors associated to an L2-acyclic group Γ an invariant P(Γ) that is a formal difference of polytopes in the vector space H1(Γ;R). This invariant is in particular defined for most 3-manifold groups, for most 2-generator 1-relator groups and for all free-by-cyclic gro…
We consider polyhedra and 4-polytopes in Minkowski spacetime - in particular, null polyhedra with zero volume, and 4-polytopes that have such polyhedra as their hyperfaces. We present the basic properties of several classes of null-faced 4-polytopes: 4-simplices, "tetrahedral diamonds" and 4-parallelotopes. We propose …
The study confirms conjectures about normals to convex polytopes in 3D space.
problem Concurrent normals problem for convex polytopes in 3D.
method Analyzes the PL concurrent normals problem for convex polytopes, proving conjectures for specific cases.
result Polytopes in 3D have points with 10 normals from interior points, confirmed for all tetrahedra and triangular prisms.
Study controllability of diffeomorphisms of simple polytopes.
problem Controllability of diffeomorphisms of simple polytopes.
method Lie group structure and controllability results for diffeomorphisms of simple polytopes.
result Identity component of the diffeomorphism group is generated by the exponential image.
The paper solves Monge-Ampère equations on reflexive polytopes, linking solvability to SYZ conjecture and tropical geometry.
problem Solvability of Monge-Ampère equations on reflexive polytopes.
method Analyzes reflexive polytopes with height functions, proving conditions for Monge-Ampère solvability and linking to SYZ conjecture.
result Conditions for Monge-Ampère solvability are necessary and sufficient, and solvability implies the SYZ conjecture for Calabi-Yau hypersurfaces.
Fast algorithm for online optimization on transport polytopes.
problem Optimizing convex objectives on transport polytopes.
method Mirror Sinkhorn algorithm combining Sinkhorn scaling and mirror descent.
result Robust and efficient online optimization for convex objectives.
Consider a Hamiltonian action of a compact Lie group on a compact symplectic manifold. A theorem of Kirwan's says that the image of the momentum mapping intersects the positive Weyl chamber in a convex polytope. I present a new proof of Kirwan's theorem, which gives explicit information on how the vertices of the polyt…
We introduce a numerical isomorphism invariant p(T) for any triangulation T of S^3. Although its definition is purely topological (inspired by the bridge number of knots), p(T) reflects the geometric properties of T. Specifically, if T is polytopal or shellable then p(T) is `small' in the sense that we obtain a linear …
In his seminal 1951 paper "Extreme forms" Coxeter \cite{cox51} observed that for n≥9 one can add vectors to the perfect lattice $\sfA_9$ so that the resulting perfect lattice, called $\sfA_9^2$ by Coxeter, has exactly the same set of minimal vectors. An inhomogeneous analog of the notion of perfect lattice is tha…
We establish geometric and topological properties of the space of value functions in finite state-action Markov decision processes. Our main contribution is the characterization of the nature of its shape: a general polytope (Aigner et al., 2010). To demonstrate this result, we exhibit several properties of the structu…
The study classifies cellular pseudomanifolds and their properties.
problem Understanding the structure of cellular pseudomanifolds.
method Analyzing the combinatorial and geometric properties of cellular pseudomanifolds.
result Complete classification of cellular pseudomanifolds with excess < 2, and progress towards excess 2.
Deep models generate geometric objects with global properties.
problem Comparing neural models' global properties from generated samples.
method Training on datasets of reflexive polytopes, comparing different representations.
result Models learn non-trivial global properties of geometric objects.
The study broadens the concept of cyclic polytopes to Veronese polytopes.
problem Extending the framework of cyclic polytopes to a broader class of polytopes.
method Described facial structure and combinatorial characterisation of facets via σ-parity alternating sequences.
result Established a bijective correspondence between combinatorial types of Veronese polytopes and partitions of finite sets.
Improved Frank-Wolfe algorithm for polytopes converges linearly with dimension dependence on optimal face.
problem Efficiently solving convex minimization problems over polytopes with linear rate.
method Revisiting Frank-Wolfe algorithm with strict complementarity assumption and away-steps.
result Linear convergence rate independent of polytope dimension for optimal face.
The paper proves consistency of archetypal analysis for multivariate data.
problem Finding optimal archetype points for multivariate data.
method Uses convex polytope to summarize data, proving consistency under specific distribution assumptions.
result Archetype points converge to optimal solution under certain conditions.
Study real projective structures on a specific Coxeter orbifold.
problem Characterize real projective structures on a noncompact Coxeter orbifold.
method Embedding and extending a Coxeter quadrilateral, perturbing to form a convex polytope, and analyzing the deformation space.
result Determine the detailed properties of the deformation space of real projective structures on the orbifold.
Given an L2-acyclic connected finite CW-complex, we define its universal L2-torsion in terms of the chain complex of its universal covering. It takes values in the weak Whitehead group Whw(G). We study its main properties such as homotopy invariance, sum formula, product formula and Poincaré d…
The paper studies deformation spaces of Coxeter truncation polytopes.
problem Understanding the geometric properties and deformations of Coxeter truncation polytopes.
method Analyzing Coxeter truncation polytopes and their deformation spaces.
result Description of deformation spaces for Coxeter truncation polytopes of dimension d⩾4. Neural networks approximate unit spheres as polytopes.
problem Approximating unit spheres with neural networks.
method Using ReLU activation in neural networks to generate polytopes.
result Neural networks can approximate unit spheres as polytopes.
The study classifies all compact hyperbolic polytopes with eight facets.
problem Classifying compact hyperbolic Coxeter four-polytopes with specific numbers of facets.
method Complete classification through mathematical analysis.
result The complete classification of compact hyperbolic Coxeter four-polytopes with eight facets.
Proves stability in Weyl polytopes using optimal transport.
problem Stability of Weyl polytopes under optimal transport.
method Optimal transport stability for reflexive Weyl polytopes.
result Weak metric SYZ conjecture holds for Delzant reflexive Weyl polytopes.
Faces of quasi-arithmetic Coxeter polytopes are also quasi-arithmetic.
problem Characterizing faces of quasi-arithmetic Coxeter polytopes.
method Proof of quasi-arithmetic property of faces and sufficient condition for arithmetic faces.
result Lower-dimensional faces of quasi-arithmetic Coxeter polytopes are quasi-arithmetic.
Given a finite collection P of convex n-polytopes in RP^n (n>1), we consider a real projective manifold M which is obtained by gluing together the polytopes in P along their facets in such a way that the union of any two adjacent polytopes sharing a common facet is convex. We prove that the real projective structure on…
This article announces the completion of the classification of rank 4 locally projective polytopes and their quotients. There are seventeen universal locally projective polytopes (nine nondegenerate). Amongst their 441 quotients are a further four (nonuniversal) regular polytopes, and 152 nonregular but section regular…
The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but non-linear and non-convex constraint. The main results of this paper are significan…
The study classifies all compact 5D polytopes with 9 facets.
problem Classifying compact hyperbolic Coxeter polytopes.
method Complete classification through mathematical analysis.
result A complete list of compact hyperbolic Coxeter 5D polytopes with 9 facets.
Smooth deformation space of Coxeter polytopes proven for orderable orbifolds.
problem Proving smoothness of deformation space for Coxeter polytopes.
method Analyzing natural map into realization space.
result Deformation space of Coxeter 3-polytopes is smooth.
New hyperbolic manifolds discovered that fiber algebraically up to dimension 8.
problem Finding hyperbolic manifolds that fiber algebraically in all dimensions 5 to 8.
method Assigning colors and states to right-angled hyperbolic polytopes and applying arguments from Jankiewicz et al.
result First examples of hyperbolic manifolds with finitely presented but not of finite type fundamental groups.
The study constructs links from polytope subgraphs and proves their hyperbolic properties.
problem Proving hyperbolic structures for links from polytope subgraphs.
method Construction of 3-manifolds from polytope subgraphs and analysis of their topology.
result Hyperbolic links are parametrized by specific subgraphs in hyperbolic polytopes.
The study classifies 331 specific 4D polytopes with 7 facets.
problem Classifying finite-volume hyperbolic Coxeter 4D polytopes.
method Complete classification through exhaustive search.
result 331 unique polytopes with 7 facets identified.
Contact manifolds' momentum polytopes are convex.
problem Understanding the structure of contact manifolds.
method Using isomorphism to toric varieties.
result Momentum polytopes of contact manifolds are convex.
Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if the set of its vertices is S\cap L where S is a sphere having no lattice points in its interior. D is called perfect if the only ellipsoid in R^n that contains S\cap L is exactly S. For a vector v of the Leech lattice Λ_{24} we define Λ_{24}(v…
Smooth approximations bound dihedral angles of convex polytopes.
problem Bounding dihedral angles of convex polytopes.
method Approximating polytopes with smooth hypersurfaces and using geometric relations.
result Established lower bounds on dihedral angles.
Polytopic Matrix Factorization models data as latent vectors from a polytope, maximizing determinant for identifiability.
problem Data decomposition with semi-structured latent vectors and polytope constraints.
method Model input data as latent vectors from a polytope, using determinant maximization for identifiability.
result Identifiability condition for polytopes with specific symmetry restrictions.
Examines nonrational polytopes and fans in toric geometry.
problem Understanding nonrational convex polytopes and fans in toric geometry.
method Discussion and interrelation of recent developments.
result Exploration of nonrational polytopes and fans in toric geometry.
The article studies factorization structures in geometry and their applications to cones and polytopes.
problem Understanding and characterizing factorization structures in geometry.
method Comprehensive study of factorization structures, including structure theory, construction of compatible polytopes and cones, and derivation of generalised Gale's evenness condition.
result Established generalised Vandermonde identities and found examples of Delzant and rational Delzant compatible polytopes.
We study finite energy classes of quasiplurisubharmonic (qpsh) functions in the setting of toric compact K{ä}hler manifolds. We characterize toric qpsh functions and give necessary and sufficient conditions for them to have finite (weighted) energy, both in terms of the associated convex function in R n , and through t…
New methods classify hyperbolic polytopes with up to 40 facets.
problem Classifying compact hyperbolic Coxeter polytopes with specific facet counts.
method New combinatorial method via point set order types.
result Proves existence of a compact hyperbolic Coxeter 29-polytope with at least 40 facets.
The paper explains how to parameterize facets of moment polytopes in real symplectic geometry.
problem Understanding the moment polytopes in real symplectic geometry.
method Parameterizing equations of facets of Delta(Z) in terms of real Ressayre's pairs of Z.
result Parameterization of facets of moment polytopes explained.
We introduce two operations named biflip and puzzle-move on simple polytopes producing polytopes with diffeomorphic moment-angle manifolds.
New algorithm samples from log-concave distributions over polytopes efficiently.
problem Sampling from log-concave distributions over polytopes.
method Generalization of Dikin walk with soft-threshold regularizer.
result Improves sampling efficiency for polytopes.