Study on volumes of random inscribed polytopes in projective geometries.
problem Estimating volumes of random inscribed polytopes in projective geometries.
method Central limit theorems and normal approximation for volumes and dual volumes of random inscribed polytopes.
result Established central limit theorems and normal approximation for volumes and dual volumes of random inscribed polytopes.
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.
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.
A planar graph is inscribable if it is combinatorial equivalent to the skeleton of a polyhedra which is inscribed in a sphere. For an inscribable graph, in its combinatorial equivalent class, if we could always find polyhedra inscribed in any given convex surface which is sufficiently close to the sphere, then we call …
Extends sphere-rhomb inscribing to more directions.
problem Bounding strictly-convex regions with rhombs inscribed in spheres.
method Combines recent work with earlier results on sphere-rhomb inscribing.
result Extends class of inscribing spheres to more directions.
Study examines Hilbert area of inscribed polygons in projective geometry.
problem Understanding Hilbert area of inscribed polygons in projective geometry.
method Examined correspondence between Fock-Goncharov and Cartesian coordinates, analyzed degeneration and Hilbert area of inscribed quadrilaterals, developed microlocal condition.
result Sequence of strictly convex domains with bounded Hilbert area and divergent Goldman parameters.
Every curve can fit countless rhombuses.
problem Finding many rhombi within any curve.
method No curve regularity assumed.
result Uncountably many rhombi fit every curve.
A regular n-gon inscribing a knot is a sequence of n points on a knot, such that the distances between adjacent points are all the same. It is shown that any smooth knot is inscribed by a regular n-gon for any n.
Study of discrete Koenigs nets and their properties.
problem Characterization and properties of discrete Koenigs nets.
method Generalization of inscribed conics to inscribed quadrics and study of Koenigs d-grids.
result Established a bijection between Koenigs d-grids and pairs of discrete autoconjugate curves.
New bounds on inscribed triangles in arbitrary planar domains.
problem Finding inscribed triangles in arbitrary planar domains with specific angle constraints.
method Proving the existence of uniformly fat triangles and not-too-fat triangles in bounded open sets.
result Existence of a maximal number Θ (between 0 and 60) for inscribed triangles with angles ≥ Θ degrees.
Derives conformal parameters of curves using inscribed circular polygons.
problem Characterizing conformal invariants of smooth curves in 3D.
method Limiting process with inscribed circular polygons, based on elementary geometry.
result Derives conformal length, curvature, and torsion via a novel method.
Paper generalizes inscribed radius estimate to hyperbolic Einstein manifolds.
problem Bounding inscribed radius in asymptotically hyperbolic Einstein manifolds.
method Generalized inscribed radius estimate to AH Einstein manifolds, combining recent work.
result Rigidity result achieved for upper bound of relative volume.
We prove a sharp estimate for the inscribed radius under certain fully nonlinear curvature flows. This estimate is asymptotically sharp on cylinders.
Square inscribed in a curve made of two graph functions.
problem Finding inscribed squares in curves formed by graph functions.
method Analysis of spectral invariants of Jordan Floer homology under curve perturbations.
result Existence of inscribed squares in curves with specific Lipschitz constants.
Study on non-orientable surfaces and inscribed rectangles in 4-manifolds.
problem Understanding non-orientable surfaces and their inscribed rectangles.
method Analyzing smooth and locally-flat non-orientable surfaces in 4-ball with specific knots, comparing results.
result Established differences between smooth and locally-flat non-orientable 4-genus of torus knots.
Smooth knots with odd Conway polynomial terms have inscribed trefoils.
problem Finding inscribed trefoils for smooth knots with specific polynomial terms.
method Using a perturbation of the double-cover of the orientation class and analyzing planar configurations.
result Smooth knots with odd quadratic terms of the Conway polynomial have inscribed trefoils.
Similar simplices can be inscribed in most smoothly embedded spheres.
problem Inscribing families of similar simplices in spheres.
method Diffeomorphic mapping and techniques from previous work on inscribing triangles.
result A dense family of spheres allows inscribing similar simplices of every pose.
We study convex polyhedra in RP3 with all their vertices on a sphere. We do not require, in particular, that the polyhedra lie in the interior of the sphere, hence the term "weakly inscribed". Such polyhedra can be interpreted as ideal polyhedra, if we regard RP3 as a combinati…
We study convex polyhedra in three-space that are inscribed in a quadric surface. Up to projective transformations, there are three such surfaces: the sphere, the hyperboloid, and the cylinder. Our main result is that a planar graph Γ is realized as the 1-skeleton of a polyhedron inscribed in the hyperboloid or cyl…
Sharp upper bounds on inscribed radius for metric spaces with convex boundary.
problem Bounding inscribed radius in metric measure spaces with convex boundary.
method Proves sharp upper bounds on inscribed radius for subsets with convex boundary.
result Sharp upper bounds on inscribed radius for subsets with convex boundary.
We prove that any cyclic quadrilateral can be inscribed in any closed convex C1-curve. The smoothness condition is not required if the quadrilateral is a rectangle.
We prove that for every smooth Jordan curve γ, if X is the set of all r∈[0,1] so that there is an inscribed rectangle in γ of aspect ratio tan(r⋅π/4), then the Lebesgue measure of X is at least 1/3. To do this, we study sets of disjoint homologically nontrivial projective planes smoothly embedde…
Floer homology applied to inscribing rectangles into curves.
problem Determining if a Jordan curve can inscribe a square.
method Constructing Floer homology from inscribed rectangles and using spectral invariants.
result A Jordan curve inscribes a square if its enclosed area exceeds half a circle's area.
We show that for every positive integer n there is a simple closed curve in the plane (which can be taken infinitely differentiable and convex) which has exactly n inscribed squares.
We consider a family of embedded, mean convex hypersurfaces which evolve by the mean curvature flow. It follows from general results of White that the inscribed radius at each point on the surface is at least Hc, where c is a constant that depends only on the initial data. Andrews recently gave a new proof…
Square can fit inside curves close to smooth ones.
problem Finding inscribed squares in nearly smooth curves.
method Using curvature and a map to relate curves, proving existence of inscribed squares.
result Curves close to smooth ones contain inscribed squares.
Square-like quadrilaterals inscribed in space curves proven for finite total curvature.
problem Square-peg problem in space curves.
method Local curvature analysis and limiting argument on approximating curves.
result Every embedded curve with finite total curvature has an inscribed square-like quadrilateral.
The abstract proves polygon inscriptions in curves with specific edge ratios.
problem Proving the existence of polygons inscribed in Jordan curves with prescribed edge ratios.
method Using the properties of differentiable curves and proportional side lengths.
result Existence of polygons inscribed in Jordan curves with prescribed edge ratios.
Bayesian optimization adapted for discrete spaces using random mappings.
problem Global optimization of expensive black-box functions with discrete variables.
method Embeds discrete space into a convex polytope, performs optimization in continuous space.
result Method outperforms existing methods in large combinatorial spaces.
The abstract shows how embeddings inscribe trapezoids or map three points to a line, proving nonexistence of certain maps.
problem Proving the nonexistence of affinely 3-regular maps in infinitely many dimensions.
method Elementary proof using embeddings and nonsingular bilinear maps.
result Recovery of nonexistence results for affinely 3-regular maps without complex algebraic techniques.
We find bounds on the difference between the writhing number of a smooth curve, and the writhing number of a polygon inscribed within. The proof is based on an extension of Fuller's difference of writhe formula to the case of polygonal curves. The results establish error bounds useful in the computation of writhe.
Fast algorithm samples confined polygons efficiently.
problem Sampling confined random equilateral closed polygons efficiently.
method Uses symplectic geometry to sample moment polytope, leading to a linear-time algorithm.
result Explicit formulas for expected distances and total curvature of vertices to the origin.
We introduce the geodesic walk for sampling Riemannian manifolds and apply it to the problem of generating uniform random points from polytopes in R^n specified by m inequalities. The walk is a discrete-time simulation of a stochastic differential equation (SDE) on the Riemannian manifold equipped with the metric induc…
We present an affine-invariant random walk for drawing uniform random samples from a convex body K⊂Rn that uses maximum volume inscribed ellipsoids, known as John's ellipsoids, for the proposal distribution. Our algorithm makes steps using uniform sampling from the John's ellipsoid of the …
In a recent paper, Brendle proved that the inscribed radius of closed embedded mean convex hypersurfaces moving by mean curvature flow is at least 1/((1+δ)H) at all points with H > C(δ,M_0). In this note, we give a shorter proof of Brendle's estimate, and of a more general result for alpha-Andrews flows, based on our r…
The paper proves that any smooth curve can have two similar inscribed rectangles.
problem Finding two similar inscribed rectangles in a smooth Jordan curve.
method Lagrangian Floer homology and differential topological computation.
result Generic doubling of inscribed rectangles in smooth Jordan curves.
Curves inscribe rectangles with positive area.
problem Finding angles for inscribing rectangles within Jordan curves.
method Proving existence of a subset of angles with measure at least A/R^2.
result Angles inscribing rectangles have measure at least A/R^2.
Continuous curves inscribe isosceles trapezoids in complex plane.
problem Proving periodic curves inscribe isosceles trapezoids.
method Lagrangian intersection problem and convergence argument.
result Continuous curves inscribe trapezoids with any similarity type.
Sparse neural encoding can store more memories as targets become sparser.
problem Storing sparse input-target associations in neural networks.
method Mathematical proofs using properties of random polytopes and sub-gaussian random vector variables.
result The capacity of neural maps increases with sparsity in target layers.
Least Squares Estimators are suboptimal for 5D convex functions.
problem Suboptimality of Least Squares Estimators in estimating multidimensional convex functions.
method Analysis of natural subclasses of convex functions in random and fixed design settings.
result Risk of LSE is n−2/d while minimax risk is n−4/(d+4) for d≥5. Paper studies inscribed sphere and lines through centers of Apollonius spheres in n dimensions.
problem Tangency of spheres and lines through their centers.
method Lie sphere geometry and two-step construction of Apollonius spheres.
result Center of inscribed sphere coincides with point PX. We prove a transversality "lifting property" for compactified configuration spaces as an application of the multijet transversality theorem: the submanifold of configurations of points on an arbitrary submanifold of Euclidean space may be made transverse to any submanifold of the configuration space of points in Euclid…
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.
We study complete minimal graphs in HxR, which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon Γ in H. We give necessary and sufficient conditions on the "lenghts" of the sides of the polygon (and all inscribed polygons in Γ) that ensure the existence…
First, we prove a special case of Knaster's problem, implying that each symmetric convex body in R^3 admits an inscribed cube. We deduce it from a theorem in equivariant topology, which says that there is no S_4-equivariant map from SO(3) to S^2, where S_4 acts on SO(3) as the rotation group of the cube and on S^2 as t…
We consider a family of embedded, mean convex hypersurfaces in a Riemannian manifold which evolve by the mean curvature flow. We show that, given any number T>0 and any δ>0, we can find a constant C0 with the following property: if t∈[0,T) and p is a point on Mt where the curvature is greater than $C_…
The paper solves a financial mathematics problem using polytopes and probability measures.
problem Maximizing the expectation of functions on probability measures.
method Identifying specific functions and using polytopes to find optimal probability measures.
result The supervertex and subvertex of polytopes maximize or minimize the expected value of certain functions.
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.