arXiv research

A number of results for C$^2$-smooth surfaces of constant width in Euclidean 3-space ${\mathbb{E}}^3$ are obtained. In particular, an integral inequality for constant width surfaces is established. This is used to prove that the ratio of volume to cubed width of a constant width surface is reduced by shrinking it along…

The paper explores properties of curves and surfaces of constant width using Fourier series.

New insights into $p$-widths of surfaces, proving optimality and calculating constants.

Find simple geodesics in hyperbolic surfaces with bounded diameter.

We give a simple procedure to estimate the smallest Lipshitz constant of a degree 1 map from a Riemannian 2-sphere to the unit 2-sphere, up to a factor of 10. Using this procedure, we are able to prove several inequalities involving this Lipshitz constant. For instance, if the smallest Lipshitz constant is at least 1, …

Given a 2-dimensional surface M and a constant C we construct a Riemannian metric g, so that diameter diam(M,g)=1 and every 1-cycle dividing M into two regions of equal area has length >C. It follows that there exists no universal inequality bounding 1-width of M in terms of its diameter. This answers a question of Ste…

Hyperbolic 3-manifolds have bounded treewidth related to their volume.

The width of a closed convex subset of Euclidean space is the distance between two parallel supporting planes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still open in dimension n > 2. In this paper we describe a necessary condition that th…

We present an alternative proof of the following fact: the hyperspace of compact closed subsets of constant width in $\mathbb R^n$ is a contractible Hilbert cube manifold. The proof also works for certain subspaces of compact convex sets of constant width as well as for the pairs of compact convex sets of constant rela…

We prove that among all constant width bodies of revolution, the minimum of the ratio of the volume to the cubed width is attained by the constant width body obtained by rotation of the Reuleaux triangle about an axis of symmetry.

In 1926 S. Nakajima (= A. Matsumura) showed that any convex body in $\R^3$ with constant width, constant brightness, and boundary of class $C^2$ is a ball. We show that the regularity assumption on the boundary is unnecessary, so that balls are the only convex bodies of constant width and brightness.

Riemannian manifolds with bounded Ricci curvature have finite Uryson width.

Let n be a natural number equal or greater than 2. In this paper we study the topological structure of certain hyperspaces of convex subsets of constant width, equipped with the Hausdorff metric topology. We focus our attention on the hyperspace cw_D(R^n) of all compact convex subsets with constant width d\in D, where …

We study surface knots in 4-space by using generic planar projections. These projections have fold points and cusps as their singularities and the image of the singular point set divides the plane into several regions. The width (or the total width) of a surface knot is a numerical invariant related to the number of po…

Study on spherical bodies of constant width on the unit sphere, proving bounds on their relative effective radius.

Flow deforms locally convex curves to curves of constant k-order width.

3-manifolds with positive scalar curvature have controlled foliations.

Establish optimal Lipschitz lower bounds for functions on manifolds with negative curvature, revealing interplay between width, boundary area, and topology.

Consider a convex polygon P in the plane, and denote by U a homothetical copy of the vector sum of P and (-P). Then the polygon U, as unit ball, induces a norm such that, with respect to this norm, P has constant Minkowskian width. We define notions like Minkowskian curvature, evolutes and involutes for polygons of con…

Given a complete non-compact surface embedded in R^3, we consider the Dirichlet Laplacian in a layer of constant width about the surface. Using an intrinsic approach to the layer geometry, we generalise the spectral results of an original paper by Duclos et al. to the situation when the surface does not possess poles. …

Improved neural network depth-width trade-offs via dynamical systems.

Study bounds Urysohn width of manifolds under surgeries.

There are many "minimax" complexity functions in mathematics: width of a tree or a link, Heegaard genus of a 3-manifold, the Cheeger constant of a Riemannian manifold. We define such a function w, "width", on countable (or finite) groups and show w(Z^k) = k-1.

Adversarial examples in deep ReLU networks with constant depth.

New examples show high tree-width in knot diagrams.

Small ball volumes imply limited Uryson width in Riemannian manifolds.

Study on stable CMC surfaces in slabs with boundary conditions.

The paper explores curvature and minimal surfaces in normed spaces.

Study finds geodesic networks for surfaces with convex boundary.

Researchers calculate the mean width of oloid and its geometric properties.

The paper proves unique geodesics on hyperbolic surfaces and finds lower bounds.

In this paper we study properties of the area evolute (AE) and the center symmetry set (CSS) of a convex planar curve $γ$. The main tool is to define a Minkowski plane where $γ$ becomes a constant width curve. In this Minkowski plane, the CSS is the evolute of $γ$ and the AE is an involute of the CSS. We prove that the…

The paper proves rigidity theorems for area widths of Riemannian manifolds.

Study on curvature measures in non-Euclidean spaces linked to Euclidean geometry.

Develops a new theory of width for embedded circles in Riemannian manifolds.

To each knot $K\subset S^3$ one can associated its knot Floer homology $\hat{HFK}(K)$, a finitely generated bigraded abelian group. In general, the nonzero ranks of these homology groups lie on a finite number of slope one lines with respect to the bigrading. The width of the homology is, in essence, the largest horizo…

Proof confirms minimal surfaces have single instance in arbitrary dimensions.

Study on 1-Uryson width of polyhedra and their covers.

New optimizers control network width scaling, improving stability and transfer across different model sizes.

The paper extends the collar theorem to non-compact surfaces using new comparison theorems.

This paper optimizes ReLU networks for approximating Hölder continuous functions.

Defines thin position for 4-manifolds and trisections, finding key relations and diagrams.

New framework connects two neural network theories, improving finite-width approximations.

Study of manifolds with specific curvature properties using capillary surfaces.

The paper develops a theory for free boundary minimal surfaces with genus at least one.

Motivated by the theory of quantum waveguides, we investigate the spectrum of the Laplacian, subject to Dirichlet boundary conditions, in a curved strip of constant width that is defined as a tubular neighbourhood of an infinite curve in a two-dimensional Riemannian manifold. Under the assumption that the strip is asym…

We provide an upper bound for the Gromov width of compact homogeneous Hodge manifolds $(M, ω)$ with $b_2(M)=1$. As an application we obtain an upper bound on the Seshadri constant $ε(L)$ where $L$ is the ample line bundle on $M$ such that $c_1(L)=[\fracωπ]$.

The paper introduces new measures for ovals and finds geometric relations between their properties.