We investigate polyhedral -manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex {\it -Hamiltonian} if it contains the full -skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so…
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48 results for “regular polytope”
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…
Diffeomorphisms of convex polytopes form a Lie group.
problem Understanding transformations of convex polytopes.
method Forming a Lie group from diffeomorphisms of convex polytopes.
result The group of diffeomorphisms of a convex polytope is a regular Lie group.
Novel singularity models for 4D harmonic forms and spinors from polytopes.
problem Understanding harmonic forms and spinors in 4D.
method Homogeneous singularity models based on regular 4-polytopes.
result Models describe cones on the 1-skeletal of polytopes.
New method uses random convex polytopes to measure representation quality.
problem Measuring the quality of deep learning representations.
method Random Polytope Descriptor method based on random convex polytopes.
result Regularization in autoencoders can degrade out-of-distribution detection.
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 …
Proves weight polytope matches with energy vectors in toric varieties.
problem Understanding the relationship between weight polytopes and energy functionals in toric varieties.
method Combines two slope formulas of K-energy in the toric setting.
result Weight polytope of Hurwitz form matches with convex hull of characteristic vectors.
New method improves sampling from logconcave distributions truncated on polytopes.
problem Sampling from logconcave distributions with polytope constraints.
method Regularized Dikin walks, using Lewis weights.
result Improved mixing time guarantees for various distributions and polytopes.
The Wythoff construction takes a -dimensional polytope , a subset of and returns another -dimensional polytope . If is a regular polytope, then is vertex-transitive. This construction builds a large part of the Archimedean polytopes and tilings in dimension 3 and 4. We want …
Chow stability is one notion of Mumford's Geometric Invariant Theory for studying the moduli space of polarized varieties. Kapranov, Sturmfels and Zelevinsky detected that Chow stability of polarized toric varieties is determined by its inherent {\it secondary polytope}, which is a polytope whose vertices correspond to…
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.
Boundary Estimates for the Monge-Ampère Equation in the Polygons with Guillemin Boundary Conditionsmath.AP
Extends boundary estimates for Monge-Ampère equations in polygonal domains.
problem Boundary regularity for Monge-Ampère equations on convex polytopes with specific boundary conditions.
method Schauder-type techniques, inspired by Donaldson's work on the Abreu equation.
result Establishes boundary regularity result for Hölder continuous right-hand sides.
Study on Funk geometry volume growth and polytope flags, verifying conjectures.
problem Volume minimization in Funk geometry and polytopes.
method Analyzing Holmes--Thompson volume, studying asymptotics, computing coefficients.
result Second highest volume coefficient minimized by unique center point, maximized by regular polygons.
We study a generalized Abreu Equation in -dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform -stability.
Wasserstein archetypal analysis finds optimal data summaries using Wasserstein metric.
problem Finding optimal data summaries using Wasserstein metric.
method Alternative formulation of archetypal analysis based on Wasserstein metric, with regularization and gradient-based computational approach.
result Existence and consistency of solutions for the regularized problem.
Study the limiting shape of solutions to the L_p-Minkowski problem as p approaches negative infinity.
problem Understanding the limiting shape of solutions to the L_p-Minkowski problem as p → -∞.
method Group-invariant method to study the asymptotic shape of solutions.
result Existence of a solution Ω^(p) to the L_p-Minkowski problem that converges to a regular polytope T as p → -∞.
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.
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 .
Neural Polytopescs.LG
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.
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.
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…
The article studies crystallizations of small covers over simple polytopes and finds unique crystallizations for the -simplex.
problem Understanding crystallizations of small covers over simple polytopes.
method Examining crystallizations of small covers over the -simplex and prism, proving uniqueness and counting equivalence classes.
result Proves uniqueness of crystallization for over -simplex and counts equivalence classes for prism.
We discuss a PL analogue of Morse theory for PL manifolds. There are several notions of regular and critical points. A point is homologically regular if the homology does not change when passing through its level, it is strongly regular if the function can serve as one coordinate in a chart. Several criteria for strong…
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.
We prove that every complete finite-volume hyperbolic 3-manifold that is tessellated into (embedded) right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable hyperbolic 4-manifold , which is also tessellated into right-angled regular pol…
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.
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.
Proves necessity of at least log2(n) layers to compute maximum of n numbers.
problem Computing the maximum of n numbers with ReLU neural networks.
method Uses lattice polytopes and duality with Newton polytopes to prove depth lower bounds.
result Proves that log2(n) hidden layers are necessary and sufficient.
Unified cosmological and Einstein polytope theories.
problem Unified understanding of cosmological and Einstein polytope theories.
method Unified combinatorial perspective of cosmological and Einstein polytope theories.
result Unified construction of cosmological and Einstein polytope theories.
The study of symmetries in manifolds derived from colored polytopes.
problem Existence and types of symmetries in rational homology 3-spheres.
method Analysis of hyperbolic manifolds and right-angled polytopes.
result Described how to create colorings with specific symmetries.
Existence and boundary regularity away from the corners are established for two-dimensional Monge-Ampère equations on convex polytopes with Guillemin boundary conditions. An important step is to derive an expansion in terms of functions and for solutions to equations of the form $\det D^2u(x,y) = y^{-…
Efficiently projects points onto polytopes, especially useful in web-scale applications.
problem Efficiently projecting points onto polytopes in large-scale applications.
method Developed a vertex-oriented incremental algorithm for polytope projection, tailored for simplex and unit-box cut polytopes.
result Majority of projections lie on vertices of polytopes, leading to significant performance improvements.
We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative …
Geometric constraints help classify hyperbolic polytopes.
problem Classifying reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.
method Established geometric constraints on compact Coxeter polytopes in hyperbolic spaces.
result Geometric constraints are useful for classifying hyperbolic polytopes.
New noncompact Coxeter polytopes found in various dimensions.
problem Classifying and constructing noncompact hyperbolic Coxeter polytopes.
method Maximal-cusp density and noncompact analog of Bogachev-Douba-Raimbault's argument.
result Infinitely many pairwise incommensurable noncompact Coxeter polytopes in dimensions 4-9.
The regular genus of certain 4-manifolds is determined, providing new insights.
problem Determining the regular genus of higher-dimensional closed PL manifolds.
method Using crystallization graphs and combinatorial topology, the regular genus is calculated for specific manifolds.
result The regular genus of is 6, and is 16.