Euclidean Ball Research Articles

Ulam floating bodies

We study a new construction of bodies from a given convex body in R n which are isomorphic to (weighted) floating bodies. We establish several properties of this new construction, including its relation to p-affine surface areas. We show that these bodies are related to Ulam' s long-standing floating body problem which asks whether Euclidean balls are the only bodies that can float, without turning, in any orientation.

Macroscopic analysis of determinantal random balls

We consider a collection of Euclidean random balls in $\mathbb{R}^{d}$ generated by a determinantal point process inducing inhibitory interaction into the balls. We study this model at a macroscopic level obtained by a zooming-out and three different regimes – Gaussian, Poissonian and stable – are exhibited as in the Poissonian model without interaction. This shows that the macroscopic behaviour erases the interactions induced by the determinantal point process.

An extension operator on bounded domains and applications

In this paper we study a sharp Hardy-Littlewood-Sobolev (HLS) type inequality with Riesz potential on bounded smooth domains. We obtain the inequality for a general bounded domain $\Omega$ and show that if the extension constant for $\Omega$ is strictly larger than the extension constant for the unit ball $B_1$ then extremal functions exist. Using suitable test functions we show that this criterion is satisfied by an annular domain whose hole is sufficiently small. The construction of the test functions is not based on any positive mass type theorems, neither on the nonflatness of the boundary. By using a similar choice of test functions with the Poisson-kernel-based extension operator we prove the existence of an abstract domain having zero scalar curvature and strictly larger isoperimetric constant than that of the Euclidean ball.

On Recognizing Shapes of Polytopes from Their Shadows

Let P and Q be two convex polytopes both contained in the interior of a Euclidean ball $$r\mathbf B ^{d}$$ . We prove that $$P=Q$$ provided that their sight cones from any point on the sphere $$rS^{d-1}$$ are congruent. We also prove an analogous result for spherical projections.

On types of degenerate critical points of real polynomial functions

In this paper, we consider the problem of identifying the type (local minimizer, maximizer or saddle point) of a given isolated real critical point c, which is degenerate, of a multivariate polynomial function f. To this end, we introduce the definition of faithful radius of c by means of the curve of tangency of f. We show that the type of c can be determined by the global extrema of f over the Euclidean ball centered at c with a faithful radius. We propose algorithms to compute faithful radius of c and determine its type.

Characterizations of the Euclidean Ball by Intersections with Planes and Slabs

In this article we prove two characterizations of the Euclidean ball: (i) the only convex body in such that every normal plane bisects the volume (or surface area) is the Euclidean ball, (ii) the only convex body K in such that for two suitable values the surface area of the intersection between any slab of width hi, and is a constant is the Euclidean ball.

Dirichlet boundary values on Euclidean balls with infinitely many solutions for the minimal surface system

We make systematic developments on Lawson–Osserman constructions relating to the Dirichlet problem (over unit disks) for minimal surfaces of high codimension in their 1977' Acta paper. In particular, we show the existence of boundary functions for which infinitely many analytic solutions and at least one nonsmooth Lipschitz solution exist simultaneously. This newly-discovered amusing phenomenon enriches the understanding on the Lawson–Osserman philosophy.

Volume growth and Bernstein theorems for translating solitons

In this paper, we study properties of the potential function of a translating soliton M in R n + 1 and the volume growth of the intersection of Euclidean balls with M . We show that if M is a convex translating soliton, then the infimum of the mean curvature function on M is zero, i.e., inf M ⁡ H = 0 . We give a condition to obtain the Bernstein theorem for the translating solitons. We also give an outline of a simple proof of the Bernstein theorem due to Bombieri–De Giorgi–Miranda.

On Some Problems for a Simplex and a Ball in R<sup>n</sup>

Let \(C\) be a convex body and let \(S\) be a nondegenerate simplex in \({\mathbb R}^n\). Denote by \(\tau S\) the image of \(S\) under homothety with a center of homothety in the center of gravity of \(S\) and the ratio \(\tau\). We mean by \(\xi(C;S)\) the minimal \(\tau>0\) such that \(C\) is a subset of the simplex \(\tau S\). Define \(\alpha(C;S)\) as the minimal \(\tau>0\) such that \(C\) is contained in a translate of \(\tau S\). Earlier the author has proved the equalities \(\xi(C;S)=(n+1)\max\limits_{1\leq j\leq n+1}\max\limits_{x\in C}(-\lambda_j(x))+1\) (if \(C\not\subset S\)), \(\alpha(C;S)=\sum\limits_{j=1}^{n+1} \max\limits_{x\in C} (-\lambda_j(x))+1.\)Here \(\lambda_j\) are the linear functions that are called the basic Lagrange polynomials corresponding to \(S\). The numbers \(\lambda_j(x),\ldots, \lambda_{n+1}(x)\) are the barycentric coordinates of a point \(x\in{\mathbb R}^n\). In his previous papers, the author investigated these formulae in the case when \(C\) is the \(n\)-dimensional unit cube \(Q_n=[0,1]^n\). The present paper is related to the case when \(C\) coincides with the unit Euclidean ball \(B_n=\{x: \|x\|\leq 1\},\) where \(\|x\|=\left(\sum\limits_{i=1}^n x_i^2 \right)^{1/2}.\) We establish various relations for \(\xi(B_n;S)\) and \(\alpha(B_n;S)\), as well as we give their geometric interpretation. For example, if \(\lambda_j(x)=l_{1j}x_1+\ldots+l_{nj}x_n+l_{n+1,j},\) then \(\alpha(B_n;S)=\sum\limits_{j=1}^{n+1}\left(\sum\limits_{i=1}^n l_{ij}^2\right)^{1/2}\). The minimal possible value of each characteristics \(\xi(B_n;S)\) and \(\alpha(B_n;S)\) for \(S\subset B_n\) is equal to \(n\). This value corresponds to a regular simplex inscribed into \(B_n\). Also we compare our results with those obtained in the case \(C=Q_n\).

Stability of capillary hypersurfaces in a Euclidean ball

We study the stability of capillary hypersurfaces in a unit Euclidean ball. It is proved that if the mass center of the generalized body enclosed by the immersed capillary hypersurface and the wetted part of the sphere is located at the origin, then the hypersurface is unstable. An immediate result is that all known examples except the totally geodesic ones and spherical caps are unstable.

Geometric rigidity of constant heat flow

Let $\Omega$ be a compact Riemannian manifold with smooth boundary and let $u_t$ be the solution of the heat equation on $\Omega$, having constant unit initial data $u_0=1$ and Dirichlet boundary conditions ($u_t=0$ on the boundary, at all times). If at every time $t$ the normal derivative of $u_t$ is a constant function on the boundary, we say that $\Omega$ has the {\it constant flow property}. This gives rise to an overdetermined parabolic problem, and our aim is to classify the manifolds having this property. In fact, if the metric is analytic, we prove that $\Omega$ has the constant flow property if and only if it is an {\it isoparametric tube}, that is, it is a solid tube of constant radius around a closed, smooth, minimal submanifold, with the additional property that all equidistants to the boundary (parallel hypersurfaces) are smooth and have constant mean curvature. Hence, the constant flow property can be viewed as an analytic counterpart to the isoparametric property. Finally, we relate the constant flow property with other overdetermined problems, in particular, the well-known Serrin problem on the mean-exit time function, and discuss a counterexample involving minimal free boundary immersions into Euclidean balls.

Compact four–dimensional euclidean space embedded in the universe

A compact four-dimensional manifold whose metric tensor has a positive determinant (named the "Euclid ball") is considered. The Euclid ball can be immersed in the Minkovskian space (which has the negative determinant) and can exist stably through the history of the universe. Since the Euclid ball has the same solution as the Schwarzschild black hole on its three-dimensional surface, an asymptotic observer can not distinguish them. If large fraction of whole energy of the pre-universe was encapsulated in Euclid balls, they behave as the dark matter in the current universe. Euclid balls already existed at the end of the cosmological inflation, and can have a heavy mass in this model, they can be a seed of supper-massive black-holes, which are necessary to initiate a forming of galaxies in the early universe. The $\gamma$-ray burst at early universe is also a possible signal of the Euclidean ball.

Intermediate links of plane curves

For a smooth complex curve C ⊂ ℂ2 we consider the link Lr = C ∩ ∂Br, where Br denotes an Euclidean ball of radius r > 0. We prove that the diagram Dr obtained from Lr by a complex stereographic projection satisfies χ(C ∩Br) = rot(Dr)−wr(Dr). As a consequence we show that if Dr has no negative Seifert circles and Lr is strongly quasipositive and fibered, then the Yamada–Vogel algorithm applied to Dr yields a quasipositive braid.

Fourier analysis and optimal Hardy–Adams inequalities on hyperbolic spaces of any even dimension

Recently, the second and third authors established the sharp Hardy–Adams inequalities of second order derivatives with the best constants on the hyperbolic space B4 of dimension four ([40], Adv. Math. (2017)). A key method used in dimension four in [40] is that the following Hardy operator when n=4 and k=2∫Bn|∇ku|2dx−∏i=1k(2i−1)2∫Bnu2(1−|x|2)2kdx can be decomposed exactly as the product of fractional Laplacians. However, such a decomposition in dimension four is no longer possible in the case n>4 and 2≤k<n2 and the situation here is thus considerably more difficult and complicated. Therefore, it remains open as whether a sharp Hardy–Adams inequality still holds on higher dimensional hyperbolic space Bn for n>4.The main purpose of this paper is to use a substantially new method of estimating the Hardy operator to establish the sharp Hardy–Adams inequalities on hyperbolic spaces Bn for all even dimension n and n≥4. As applications of such inequalities, we will improve substantially the known Adams inequalities on hyperbolic space Bn in the literature and also strengthen the classical Adams' inequality and the Hardy inequality on Euclidean balls in any even dimension. The later inequality can be viewed as the borderline case of the sharp Hardy–Sobolev–Maz'ya inequalities for higher order derivatives in high dimensions obtained recently by the second and third authors [41].

Covering a Ball by Smaller Balls

We prove that, for any covering of a unit d-dimensional Euclidean ball by smaller balls, the sum of radii of the balls from the covering is greater than d. We also investigate the problem of finding lower and upper bounds for the sum of powers of radii of the balls covering a unit ball.

On CMC free-boundary stable hypersurfaces in a Euclidean ball

In this note, we prove that if B is the unit ball centred in the origin in the Euclidean space with dimension $$n+1, n\ge 2$$ , then a CMC free-boundary stable hypersurface $$\Sigma $$ in B satisfies I $$\begin{aligned} \frac{nH^2}{2}\int _{\Sigma }(1-|x|^2){ dvol}_{\Sigma }+nA\le L\le nA\left( 1+H \right) , \end{aligned}$$ where L, A and H denote the length of $$\partial \Sigma $$ , the area of $$\Sigma $$ and the mean curvature of $$\Sigma $$ , respectively, and the orientation of $$\Sigma $$ is in a such way that $$H\ge 0$$ . The left side of (I) is an equality if, and only if, $$\Sigma $$ is a totally geodesic disk or a spherical cap. Consequently, if the boundary $$\partial \Sigma $$ of $$\Sigma $$ is embedded then $$\Sigma $$ must be either totally geodesic or starshaped with respect to the center of the ball. This result is a slightly improvement of a theorem proved by Ros and Vergasta. In particular, if $$n=2$$ (in this case its not necessary to assume the boundary $$\partial \Sigma $$ is embedded), the only CMC free-boundary stable surfaces in B are the totally geodesic disks or the spherical caps. This classification result was proved very recently by Nunes using an extended stability result and a modified Hersch type balancing argument to get a better control on the genus and on the number of connected components of the boundary of the surfaces. We don’t use that modified Hersch type argument. However, we use a Nunes stability type lemma and a crucial result due to Ros and Vergasta.Our technique, considering a Nunes stability type lemma, can be applied to study sets which are stable for the volume-constrained least area problem within the unit ball, and provide a proof for the Sternberg–Zumbrun’s conjecture.

Wandering domains for diffeomorphisms of the k-torus: a remark on a theorem by Norton and Sullivan

We show that there is no C^{k+1} diffeomorphism of the k-torus which is semiconjugate to a minimal translation and has a wandering domain all of whose iterates are Euclidean balls.

Eigenvalues of the Wentzell–Laplace operator and of the fourth order Steklov problems

We prove a sharp upper bound and a lower bound for the first nonzero eigenvalue of the Wentzell–Laplace operator on compact manifolds with boundary and an isoperimetric inequality for the same eigenvalue in the case where the manifold is a bounded domain in a Euclidean space. We study some fourth order Steklov problems and obtain isoperimetric upper bound for the first eigenvalue of them. We also find all the eigenvalues and eigenfunctions for two kind of fourth order Steklov problems on a Euclidean ball.

Dimension improvement in Dhar's refutation of the Eden conjecture

We consider the Eden model on the d-dimensional hypercubical unoriented lattice, for large d. Initially, every lattice point is healthy, except the origin which is infected. Then, each infected lattice point contaminates any of its neighbours with rate 1. The Eden model is equivalent to first passage percolation, with exponential passage times on edges. The Eden conjecture states that the limit shape of the Eden model is a Euclidean ball. By pushing the computations of Dhar [5] a little further with modern computers and efficient implementation we obtain improved bounds for the speed of infection. This shows that the Eden conjecture does not hold in dimension superior to 22 (the lowest known dimension was 35).

Convolution equations and mean-value theorems for solutions of linear elliptic equations with constant coefficients in the complex plane

Convolution equations generated by distributions with compact supports and the corresponding mean value theorems was investigated by many authors. In particular, Volchkov described a wide class of radial distributions with compact supports such that solutions of the corresponding convolution equations in open Euclidean balls can be efficiently characterized in terms of the Bessel functions. This characterization implies different corollaries such as uniqueness theorems for solutions of the corresponding convolution equations and two-radius theorems that go back to John (1934) and Delsarte (1961), respectively.