Euclidean Ball Research Articles

A negative answer to Ulam's Problem 19 from the Scottish Book

We give a negative answer to Ulam's Problem 19 from the Scottish Book asking is a solid of uniform density which will float in water in every position a sphere ? Assuming that the density of water is $1$, we show that there exists a strictly convex body of revolution $K\subset \mathbb{R}^3$ of uniform density $\frac{1}{2}$, which is not a Euclidean ball, yet floats in equilibrium in every orientation. We prove an analogous result in all dimensions $d\ge 3$.

On an Equichordal Property of a Pair of Convex Bodies

Let $$d\ge 2$$ and let K and L be two convex bodies in $${{\mathbb {R}}^d}$$ such that $$L\subset {\text {int}}K$$ and the boundary of L does not contain a segment. If K and L satisfy the $$(d+1)$$ -equichordal property, i.e., for any line l supporting the boundary of L and the points $$\{\zeta _{\pm }\}$$ of the intersection of the boundary of K with l, $$\begin{aligned} {\text {dist}}^{d+1}(L\cap l, \zeta _+)+{\text {dist}}^{d+1}(L\cap l, \zeta _-)=2\sigma ^{d+1} \end{aligned}$$ holds, where the constant $$\sigma $$ is independent of l, does it follow that K and L are concentric Euclidean balls? We prove that if K and L have $$C^2$$ -smooth boundaries and L is a body of revolution, then K and L are concentric Euclidean balls.

Convex floating bodies of equilibrium

We study a long standing open problem by Ulam, which is whether the Euclidean ball is the unique body of uniform density which will float in equilibrium in any direction. We answer this problem in the class of origin symmetric n n -dimensional convex bodies whose relative density to water is 1 2 \frac {1}{2} . For n = 3 n=3 , this result is due to Falconer.

Gromov–Hausdorff limits of Kähler manifolds with Ricci curvature bounded below

We show that non-collapsed Gromov–Hausdorff limits of polarized Kähler manifolds, with Ricci curvature bounded below, are normal projective varieties, and the metric singularities of the limit space are precisely given by a countable union of analytic subvarieties. This extends a fundamental result of Donaldson–Sun, in which 2-sided Ricci curvature bounds were assumed. As a basic ingredient we show that, under lower Ricci curvature bounds, almost Euclidean balls in Kähler manifolds admit good holomorphic coordinates. Further applications are integral bounds for the scalar curvature on balls, and a rigidity theorem for Kähler manifolds with almost Euclidean volume growth.

A Metric Version of Poincar\xe9\u2019s Theorem Concerning Biholomorphic Inequivalence of Domains

We show that if Y_jsubset mathbb {C}^{n_j} is a bounded strongly convex domain with C^3-boundary for j=1,dots ,q, and X_jsubset mathbb {C}^{m_j} is a bounded convex domain for j=1,ldots ,p, then the product domain prod _{j=1}^p X_jsubset mathbb {C}^m cannot be isometrically embedded into prod _{j=1}^q Y_jsubset mathbb {C}^n under the Kobayashi distance, if p>q. This result generalises Poincaré’s theorem which says that there is no biholomorphic map from the polydisc onto the Euclidean ball in mathbb {C}^n for nge 2. The method of proof only relies on the metric geometry of the spaces and will be derived from a more general result for products of proper geodesic metric spaces with the sup-metric. In fact, the main goal of the paper is to establish a general criterion, in terms of certain asymptotic geometric properties of the individual metric spaces, that yields an obstruction for the existence of an isometric embedding between product metric spaces.

On the Spectra of Three Steklov Eigenvalue Problems on Warped Product Manifolds

Let \(M^n=[0,R]\times \mathbb {S}^{n-1}\) be an n-dimensional (\(n\ge 2\)) smooth Riemannian manifold equipped with the warped product metric \(g=dr^2+h^2(r)g_{\mathbb {S}^{n-1}}\) and diffeomorphic to a Euclidean ball. Assume that M has strictly convex boundary. First, for the classical Steklov eigenvalue problem, we obtain an optimal lower (upper, respectively) bound for its spectrum in terms of \(h'(R)/h(R)\) when \(\mathrm {Ric}_g\ge 0\) (\(\le 0\), respectively). Second, for two fourth-order Steklov eigenvalue problems studied by Kuttler and Sigillito in 1968, we derive a lower bound for their spectra in terms of either \(h'(R)/h^3(R)\) or \(h'(R)/h(R)\) when \(\mathrm {Ric}_g\ge 0\), which is optimal for certain cases; in particular, we confirm a conjecture raised by Q. Wang and C. Xia for warped product manifolds of dimension \(n=2\) or \(n\ge 4\). For some proofs we utilize the Reilly’s formula and reveal a new feature on its use.

Ground states of semilinear elliptic problems with applications to the Allen\u2013Cahn equation on the sphere

We study solutions of Delta u - F'(u)=0, where the potential F can have an arbitrary number of wells at arbitrary heights, including bottomless wells with subcritical decay. In our setting, ground state solutions correspond to unstable solutions of least energy. We show that in convex domains of {mathbb {R}}^N and manifolds with {text {Ric}}ge 0, ground states are always of mountain-pass type and have Morse index 1. In addition, we prove symmetry of the ground states if the domain is either an Euclidean ball or the entire sphere S^{N}. For the Allen–Cahn equation varepsilon ^2Delta u - W'(u)=0 on S^{N}, we prove the ground state is unique up to rotations and corresponds to the equator as a minimal hypersurface. We also study bifurcation at the energy level of the ground state as varepsilon rightarrow 0, showing that the first N+1 min-max Allen–Cahn widths of S^{N} are ground states, and we prove a gap theorem for the corresponding (N+2)-th min-max solution.

Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball

In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the $(n+1)$-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for $n=2$ we obtain a Minkowski-type inequality and for $n=3$ we obtain an optimal Willmore-type inequality. To prove these estimates, we employ a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.

Homothetic packings of centrally symmetric convex bodies

A centrally symmetric convex body is a convex compact set with non-empty interior that is symmetric about the origin. Of particular interest are those that are both smooth and strictly convex—known here as regular symmetric bodies—since they retain many of the useful properties of the d-dimensional Euclidean ball. We prove that for any given regular symmetric body C, a homothetic packing of copies of C with randomly chosen radii will have a (2, 2)-sparse planar contact graph. We further prove that there exists a comeagre set of centrally symmetric convex bodies C where any (2, 2)-sparse planar graph can be realised as the contact graph of a stress-free homothetic packing of C.

Isoparametric foliations and the Pompeiu property

<abstract><p>A bounded domain $ \Omega $ in a Riemannian manifold $ M $ is said to have the Pompeiu property if the only continuous function which integrates to zero on $ \Omega $ and on all its congruent images is the zero function. In some respects, the Pompeiu property can be viewed as an overdetermined problem, given its relation with the Schiffer problem. It is well-known that every Euclidean ball fails to have the Pompeiu property while spherical balls have the property for almost all radii (Ungar's Freak theorem). In the present paper we discuss the Pompeiu property when $ M $ is compact and admits an isoparametric foliation. In particular, we identify precise conditions on the spectrum of the Laplacian on $ M $ under which the level domains of an isoparametric function fail to have the Pompeiu property. Specific calculations are carried out when the ambient manifold is the round sphere, and some consequences are derived. Moreover, a detailed discussion of Ungar's Freak theorem and its generalizations is also carried out.</p></abstract>

Geometric systems of unbiased representatives

Let P be a finite point set in Rd, B be a bicoloring of P and O be a family of geometric objects (that is, intervals, boxes, balls, etc). An object from O is called balanced with respect to B if it contains the same number of points from each color of B. For a collection B of bicolorings of P, a geometric system of unbiased representatives (G-SUR) is a subset O′⊆O such that for any bicoloring B of B there is an object in O′ that is balanced with respect to B.We pose and study problems on finding G-SURs. We obtain general bounds on the size of G-SURs consisting of intervals, size-restricted intervals, axis-parallel boxes and Euclidean balls. We show that the G-SUR problem is NP-Hard even in the simple case of points on a line and interval ranges. Furthermore, we study a related problem on determining the size of the largest and smallest balanced intervals for points on the real line with a random distribution and coloring.Our results are a natural extension to a geometric context of the work initiated by Balachandran et al. (Discrete Mathematics, 2018) on arbitrary systems of unbiased representatives.

Central limit theorems for stochastic wave equations in dimensions one and two

Fix $$d\in \{1,2\}$$ , we consider a d-dimensional stochastic wave equation driven by a Gaussian noise, which is temporally white and colored in space such that the spatial correlation function is integrable and satisfies Dalang’s condition. In this setting, we provide quantitative central limit theorems for the spatial average of the solution over a Euclidean ball, as the radius of the ball diverges to infinity. We also establish functional central limit theorems. A fundamental ingredient in our analysis is the pointwise $$L^p$$ -estimate for the Malliavin derivative of the solution, which is of independent interest. This paper is another addendum to the recent research line of averaging stochastic partial differential equations.

A solution to the fifth and the eighth Busemann-Petty problems in a small neighborhood of the Euclidean ball

We show that the fifth and the eighth Busemann-Petty problems have positive solutions for bodies that are sufficiently close to the Euclidean ball in the Banach-Mazur distance.

LPV control and analysis under uncertain initial conditions

This paper is focused on the control of discrete-time linear parameter-varying (LPV) systems with uncertain initial conditions. The synthesis objective is to design a feedback LPV controller that ensures that the closed-loop system is asymptotically stable and some performance inequality is satisfied for all permissible parameter trajectories. The performance measure used is similar to the ℓ2-induced norm except that the initial state and the disturbance input are considered to reside in two separate norm balls; that is, the supremum of the ℓ2-norm of the performance output is minimized subject to the initial state taking values in a unit Euclidean ball and the disturbance input taking values in a unit ℓ2-norm ball. The paper provides convex analysis and synthesis conditions for solving this problem and demonstrates that using nonstationary LPV controllers instead of standard LPV ones could result in a significantly improved performance. The paper also shows how to determine bounding envelopes for the performance output using the analysis result. An illustrative example concludes the paper.

A cookbook for approximating Euclidean balls and for quadrature rules in finite element methods for nonlocal problems

The implementation of finite element methods (FEMs) for nonlocal models with a finite range of interaction poses challenges not faced in the partial differential equations (PDEs) setting. For example, one has to deal with weak forms involving double integrals which lead to discrete systems having higher assembly and solving costs due to possibly much lower sparsity compared to that of FEMs for PDEs. In addition, one may encounter nonsmooth integrands. In many nonlocal models, nonlocal interactions are limited to bounded neighborhoods that are ubiquitously chosen to be Euclidean balls, resulting in the challenge of dealing with intersections of such balls with the finite elements. We focus on developing recipes for the efficient assembly of FEM stiffness matrices and on the choice of quadrature rules for the double integrals that contribute to the assembly efficiency and also posses sufficient accuracy. A major feature of our recipes is the use of approximate balls, e.g. several polygonal approximations of Euclidean balls, that, among other advantages, mitigate the challenge of dealing with ball-element intersections. We provide numerical illustrations of the relative accuracy and efficiency of the several approaches we develop.

On Properties of a Regular Simplex Inscribed into a Ball

Let $B$ be a Euclidean ball in ${\mathbb R}^n$ and let $C(B)$ be a space of continuos functions $f:B\to{\mathbb R}$ with the uniform norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ we mean a set of polynomials of degree $\leq 1$, i.e., a set of linear functions upon ${\mathbb R}^n$. The interpolation projector $P:C(B)\to \Pi_1({\mathbb R}^n)$ with the nodes $x^{(j)}\in B$ is defined by the equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$, $j=1,\ldots, n+1$.The norm of $P$ as an operator from $C(B)$ to $C(B)$ can be calculated by the formula $\|P\|_B=\max_{x\in B}\sum |\lambda_j(x)|.$ Here $\lambda_j$ are the basic Lagrange polynomials corresponding to the $n$-dimensional nondegenerate simplex $S$ with the vertices $x^{(j)}$. Let $P^\prime$ be a projector having the nodes in the vertices of a regular simplex inscribed into the ball. We describe the points $y\in B$ with the property $\|P^\prime\|_B=\sum |\lambda_j(y)|$. Also we formulate some geometric conjecture which implies that $\|P^\prime\|_B$ is equal to the minimal norm of an interpolation projector with nodes in $B$. We prove that this conjecture holds true at least for $n=1,2,3,4$.

On mixed Hodge–Riemann relations for translation-invariant valuations and Aleksandrov–Fenchel inequalities

A version of the Hodge–Riemann relations for valuations was recently conjectured and proved in several special cases by [J. Kotrbatý, On Hodge–Riemann relations for translation-invariant valuations, preprint (2020), arXiv:2009.00310]. The Lefschetz operator considered there arises as either the product or the convolution with the mixed volume of several Euclidean balls. Here we prove that in (co-)degree one, the Hodge–Riemann relations persist if the balls are replaced by several different (centrally symmetric) convex bodies with smooth boundary with positive Gauss curvature. While these mixed Hodge–Riemann relations for the convolution directly imply the Aleksandrov–Fenchel inequality, they yield for the dual operation of the product a new inequality. This new inequality strengthens classical consequences of the Aleksandrov–Fenchel inequality for lower-dimensional convex bodies and generalizes some of the geometric inequalities recently discovered by [S. Alesker, Kotrbatý’s theorem on valuations and geometric inequalities for convex bodies, preprint (2020), arXiv:2010.01859].

Some characterizations of the Euclidean ball

Some characterizations of the Euclidean ball

Nonnegative Blind Source Separation for Ill-Conditioned Mixtures via John Ellipsoid.

Nonnegative blind source separation (nBSS) is often a challenging inverse problem, namely, when the mixing system is ill-conditioned. In this work, we focus on an important nBSS instance, known as hyperspectral unmixing (HU) in remote sensing. HU is a matrix factorization problem aimed at factoring the so-called endmember matrix, holding the material hyperspectral signatures, and the abundance matrix, holding the material fractions at each image pixel. The hyperspectral signatures are usually highly correlated, leading to a fast decay of the singular values (and, hence, high condition number) of the endmember matrix, so HU often introduces an ill-conditioned nBSS scenario. We introduce a new theoretical framework to attack such tough scenarios via the John ellipsoid (JE) in functional analysis. The idea is to identify the maximum volume ellipsoid inscribed in the data convex hull, followed by affinely mapping such ellipsoid into a Euclidean ball. By applying the same affine mapping to the data mixtures, we prove that the endmember matrix associated with the mapped data has condition number 1, the lowest possible, and that these (preconditioned) endmembers form a regular simplex. Exploiting this regular structure, we design a novel nBSS criterion with a provable identifiability guarantee and devise an algorithm to realize the criterion. Moreover, for the first time, the optimization problem for computing JE is exactly solved for a large-scale instance; our solver employs a split augmented Lagrangian shrinkage algorithm with all proximal operators solved by closed-form solutions. The competitiveness of the proposed method is illustrated by numerical simulations and real data experiments.

A new look at random projections of the cube and general product measures

A consequence of the celebrated Dvoretzky–Milman theorem is a strong law of large numbers for d-dimensional random projections of the n-dimensional cube. It shows that, with respect to the Hausdorff distance, a uniform random projection of the cube [−1/n,+1/n]n onto Rd converges almost surely to a centered d-dimensional Euclidean ball of radius 2/π, as n→∞. We start by providing an alternative proof of this strong law via the Artstein–Vitale law of large numbers for random compact sets. Then, for every point inside the ball of radius 2/π, we determine the asymptotic number of vertices and the volume of the part of the cube projected ‘close’ to this point. More generally, we study large deviations for random projections of arbitrary product measures. Let ν⊗n be the n-fold product measure of a Borel probability measure ν on R, and let I be uniformly distributed on the Stiefel manifold of orthogonal d-frames in Rn. It is shown that the sequence of random measures ν⊗n∘(n−1/2I∗)−1, n∈N, satisfies a large deviation principle with probability 1. The rate function is explicitly identified in terms of the moment generating function of ν. At the heart of the proofs lies a transition trick which allows to replace the uniform projection by the Gaussian one. A number of concrete examples are discussed as well, including the uniform distributions on the cube [−1,1]n and the discrete cube {−1,1}n as special cases.