arXiv research

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

Study isoperimetric equalities for special closed curves.

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…

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…

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 …

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

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

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.

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

Riemannian manifolds with bounded Ricci curvature have finite Uryson width.

This paper focuses on curves and surfaces of constant width, with some additional results about general ovals. We emphasize the use of Fourier series to derive properties, some of which are known. Amongst other results, we show that the perimeter of an oval is $π$ times its average width, and provide a bound for the ra…

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 explores connections between perimeter, area, and visual angle of convex sets.

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

Quantitative CLTs show neural network distributions converge to Gaussian as width increases.

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

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…

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…

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

Study bounds Urysohn width of manifolds under surgeries.

Study on Gaussian-width complexity on statistical manifolds and its applications in learning and recovery.

This paper optimizes ReLU networks for approximating Hölder continuous functions.

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.

The paper addresses the gap between theoretical and practical confidence set widths in universal inference.

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

If $(M^n, g)$ is a closed Riemannian manifold where every unit ball has volume at most $ε_n$ (a sufficiently small constant), then the $(n-1)$-dimensional Uryson width of $(M^n, g)$ is at most 1.

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

We prove that a bounded open set U in Euclidean n-space has k-width less than C(n) Volume(U)^{k/n}. Using this estimate, we give lower bounds for the k-dilation of degree 1 maps between certain domains in Euclidean space. In particular, we estimate the smallest (n-1)-dilation of any degree 1 map between two n-dimension…

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

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

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.

Uniform diameter bound for reflection group disk patterns.

Deep networks can be biased to learn top eigenfunctions of the kernel outside the training set.

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, …

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

Logarithmic network width suffices for robust memorization.

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

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.

This work analyzes how deep neural networks' expressiveness increases with depth and width.

Sine activation functions enable two-layer neural networks to learn modular addition more efficiently.

This paper tightens bounds on the smallest eigenvalue of NTK for deep ReLU networks.

Wide neural networks become linear, with constant tangent kernel, due to Hessian scaling.

New framework for understanding infinite-width neural networks.

The paper establishes new band width inequalities for Riemannian bands.