arXiv research

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

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

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…

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

Study isoperimetric equalities for special closed curves.

Given a Riemannian metric on a homotopy $n$-sphere, sweep it out by a continuous one-parameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as possible yet preserving the sweepout. We show: Each curve in the tightened sweepout …

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

The paper explores cycloids in normed planes, deriving properties of closed curves.

Abstract: Comprehensive overview on curvatures in normed planes.

Find simple geodesics in hyperbolic surfaces with bounded diameter.

Paper improves isoperimetric inequality with new curve properties.

Study on width of Jordan curves in complex projective space, distinguishing quasicircles.

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…

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, …

Wide ribbons of large width can connect knots of different types.

For negatively curved manifolds, a condition is found for intrinsic ultracontractivity of heat semigroups.

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 …

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…

A simple closed curve $γ$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(γ)-2δ(γ)=3$ for anti-convex curves, where $i(γ)$ is the number of independent (true…

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

The paper finds lower bounds on curve length with constraints on width and inradius.

The paper bounds the min-max width of embedded circles on spheres and manifolds.

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…

Study on Gehring link problem and width of bands in curved manifolds.

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.

Small ball volumes imply limited Uryson width in Riemannian manifolds.

Study finds geodesic networks for surfaces with convex boundary.

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

The study reveals a transition in neural network performance from infinite-width to variance-limited behavior as dataset size increases.

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

Lower bound for Steklov eigenvalues on negatively curved manifolds.

This paper optimizes ReLU networks for approximating Hölder continuous functions.

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

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 $c_1(L)=[\fracωπ]$.

This paper studies bounds for the Lipschitz constant of random neural networks.

We investigate several integer invariants of curves in 3-space. We demonstrate relationships of these invariants to crossing number and to total curvature.

Study compares hyperbolic and extremal lengths for shortest curves.

Logarithmic network width suffices for robust memorization.

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…

Fisher width is a geometric measure of complexity on statistical manifolds.

Wider neural networks perform better than deeper ones with the same number of parameters.