Euclidean Unit Sphere Research Articles

Transport-entropy and functional forms of Blaschke–Santaló inequalities

We explore alternative functional or transport-entropy formulations of the Blaschke–Santaló inequality. In particular, we obtain new Blaschke–Santaló inequalities for s -concave functions. We also obtain new sharp symmetrized transport-entropy inequalities for a large class of spherically invariant probability measures, including the uniform measure on the unit Euclidean sphere and generalized Cauchy and Barenblatt distributions.

The Minimal Spherical Dispersion

We prove upper and lower bounds on the minimal spherical dispersion, improving upon previous estimates obtained by Rote and Tichy in (Anz Österreich Akad Wiss Math Nat Kl 132:3–10, 1995). In particular, we see that the inverse N(ε,d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N(\\varepsilon ,d)$$\\end{document} of the minimal spherical dispersion is, for fixed ε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon >0$$\\end{document}, linear in the dimension d of the ambient space. We also derive upper and lower bounds on the expected dispersion for points chosen independently and uniformly at random from the Euclidean unit sphere. In terms of the corresponding inverse N~(ε,d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{N}(\\varepsilon ,d)$$\\end{document}, our bounds are optimal with respect to the dependence on ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}.

The region of the unit Euclidean sphere that admits a class of (r, s)-linear Weingarten hypersurfaces

In the unit Euclidean sphere Sn+1, we deal with a class of hypersurfaces that were characterized in [23] as the critical points of a variational problem, the so-called (r, s)-linear Weingarten hypersurfaces (0 ≤ r ≤s ≤ n−1); namely, the hypersurfaces of Sn+1 that has a linear combination arHr+1+...+asHs+1 of their higher order mean curvatures Hr+1 and Hs+1 being a real constant, where ar,..., ar are nonnegative real numbers (with at least one non zero). By assuming a geometric constraint involving the higher order mean curvatures of these hypersurfaces, we prove a uniqueness result for strongly stable (r, s)-linear Weingarten hypersurfaces immersed in a certain region determined by a geodesic sphere of Sn+1. We also establish a nonexistence result in another region of Sn+1 for strongly stable Weingarten (r, s)-linear hypersurfaces.

Asymptotic expected T‐functionals of random polytopes with applications to Lp$L_p$ surface areas

AbstractAn asymptotic formula is proved for the expected T‐functional of the convex hull of independent and identically distributed random points sampled from the Euclidean unit sphere in according to an arbitrary positive continuous density. As an application, the approximation of the sphere by random polytopes in terms of surface area differences is discussed.

On the index of stability of (r, s)-linear Weingarten Clifford tori

For entire numbers r and s satisfying 0 ≤ r ≤ s ≤ n − 2, we showed that the index of (r, s)-stability of a (r, s)-linear Weingarten Clifford torus immersed into the (n + 1)-dimensional unit Euclidean sphere, that has a linear combination of their higher order mean curvatures Hr+1 and Hs+1 being null, is exactly equal to n + 3 provided that a geometric condition involving Hr+2 and Hs+2 is satisfied.

Spherical centroid bodies

The spherical centroid body of a centrally-symmetric convex body in the Euclidean unit sphere is introduced. Two alternative definitions---one geometric, the other probabilistic in nature---are given and shown to lead to the same objects. The geometric approach is then used to establish a number of basic properties of spherical centroid bodies, while the probabilistic approach inspires the proof of a spherical analogue of the classical polar Busemann--Petty centroid inequality.

Laplace eigenvalues of ellipsoids obtained as analytic perturbations of the unit sphere

We study the eigenvalues of the Laplacian on ellipsoids that are obtained as perturbations of the standard Euclidean unit sphere in dimension two. A comparison of these eigenvalues with those of the standard Euclidean unit sphere is obtained under a Gaussian curvature condition, in line with the Lichnerowicz theorem on the first positive eigenvalue on a compact Riemannian manifold.

On the relativistic flocks over the unit sphere and the hyperboloid in a bonding force field

We study emergent collective dynamics for the relativistic Cucker–Smale (RCS) model in a bonding force field on an abstract Riemannian manifold. The abstract RCS model in a bonding force field contains forcing terms involved with geometric quantities, such as parallel transport, Riemannian metric tensor, and logarithm mapping on manifolds. We consider two explicit realizations of the RCS model on the Euclidean unit sphere and the hyperboloid and present refined emergent dynamics of the explicit RCS models and asymptotic behaviors. We also show that the explicit RCS models reduce to the relativistic Kuramoto-type models with a memory effect for a one-dimensional setting.

Optimization on the Euclidean Unit Sphere

We consider the problem of minimizing a continuously differentiable function f of m linear forms in n variables on the Euclidean unit sphere. We show that this problem is equivalent to minimizing the same function of related m linear forms (but now in m variables) on the Euclidean unit ball. When the linear forms are known, this results in a drastic reduction in problem size whenever m ≪ n and allows to solve potentially large scale non-convex such problems. We also provide a test to detect when a polynomial is a polynomial in a fixed number of forms. Finally, we identify two classes of functions with no spurious local minima on the sphere: (i) quasi-convex polynomials of odd degree and (ii) nonnegative and homogeneous functions. Finally, odd degreed forms have only nonpositive local minima and at most (d − 1) m are strictly negative.

Asymptotic interplay of states and adaptive coupling gains in the Lohe Hermitian sphere model

<p style='text-indent:20px;'>We study emergent dynamics of the Lohe Hermitian sphere (LHS) model with the same free flows under the dynamic interplay between state evolution and adaptive couplings. The LHS model is a complex counterpart of the Lohe sphere (LS) model on the unit sphere in Euclidean space, and when particles lie in the Euclidean unit sphere embedded in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb C^{d+1} $\end{document}</tex-math></inline-formula>, it reduces to the Lohe sphere model. In the absence of interactions between states and coupling gains, emergent dynamics have been addressed in [<xref ref-type="bibr" rid="b23">23</xref>]. In this paper, we further extend earlier results in the aforementioned work to the setting in which the state and coupling gains are dynamically interrelated via two types of coupling laws, namely anti-Hebbian and Hebbian coupling laws. In each case, we present two sufficient frameworks leading to complete aggregation depending on the coupling laws, when the corresponding free flow is the same for all particles.</p>

Index estimates for closed minimal submanifolds of the sphere

In this paper we are interested in comparing the spectra of two elliptic operators acting on a closed minimal submanifold of the Euclidean unit sphere. Using an approach introduced by Savo in [A Savo. Index Bounds for Minimal Hypersurfaces of the Sphere. Indiana Univ. Math. J. 59 (2010), 823-837.], we are able to compare the eigenvalues of the stability operator acting on sections of the normal bundle and the Hodge Laplacian operator acting on $1$-forms. As a byproduct of the technique and under a suitable hypothesis on the Ricci curvature of the submanifold we obtain that its first Betti's number is bounded from above by a multiple of the Morse index, which provide evidence for a well-known conjecture of Schoen and Marques & Neves in the setting of higher codimension.

Homogeneous polynomials and spurious local minima on the unit sphere

We consider forms on the Euclidean unit sphere. We obtain a simple and complete characterization of all points that satisfies the standard second-order necessary condition of optimality. It is stated solely in terms of the value of (i) f, (ii) the norm of its gradient, and (iii) the first two smallest eigenvalues of its Hessian, all evaluated at the point. In fact this property also holds for twice continuous differentiable functions that are positively homogeneous. We also characterize a class of degree-d forms with no spurious local minima on \(\mathbb {S}^{n-1}\) by using a property of gradient ideals in algebraic geometry.

ON Free Boundary Minimal Hypersurfaces in the Riemannian Schwarzschild Space

In contrast with the three-dimensional case (cf. Montezuma in Bull Braz Math Soc), where rotationally symmetric totally geodesic free boundary minimal surfaces have Morse index one; we prove in this work that the Morse index of a free boundary rotationally symmetric totally geodesic hypersurface of the n-dimensional Riemannnian Schwarzschild space with respect to variations that are tangential along the horizon is zero, for \(n\ge 4\). Moreover, we show that there exist non-compact free boundary minimal hypersurfaces which are not totally geodesic, \(n\ge 8\), with Morse index equal to 0. In addition, it is shown that, for \(n\ge 4\), there exist infinitely many non-compact free boundary minimal hypersurfaces, which are not congruent to each other, with infinite Morse index. We also study the density at infinity of a free boundary minimal hypersurface with respect to a minimal cone constructed over a minimal hypersurface of the unit Euclidean sphere. We obtain a lower bound for the density in terms of the area of the boundary of the hypersurface and the area of the minimal hypersurface in the unit sphere. This lower bound is optimal in the sense that only minimal cones achieve it.

Best $$L^{2}$$-Extension of Algebraic Polynomials from the Unit Euclidean Sphere to a Concentric Sphere

Best $$L^{2}$$-Extension of Algebraic Polynomials from the Unit Euclidean Sphere to a Concentric Sphere

Non-Gaussian hyperplane tessellations and robust one-bit compressed sensing

We show that a tessellation generated by a small number of random affine hyperplanes can be used to approximate Euclidean distances between any two points in an arbitrary bounded set T , where the random hyperplanes are generated by subgaussian or heavy-tailed normal vectors and uniformly distributed shifts. The number of hyperplanes needed for constructing such tessellations is determined by natural metric complexity measures of the set T and the wanted approximation error. In comparison, previous results in this direction were restricted to Gaussian hyperplane tessellations of subsets of the Euclidean unit sphere. As an application, we obtain new reconstruction results in memoryless one-bit compressed sensing with non-Gaussian measurement matrices: by quantizing at uniformly distributed thresholds, it is possible to accurately reconstruct low-complexity signals from a small number of one-bit quantized measurements, even if the measurement vectors are drawn from a heavy-tailed distribution. These reconstruction results are uniform in nature and robust in the presence of pre-quantization noise on the analog measurements as well as adversarial bit corruptions in the quantization process. Moreover, if the measurement matrix is subgaussian then accurate recovery can be achieved via a convex program.

A Simons type integral inequality for closed submanifolds in the product space [formula omitted

A Simons type formula for submanifolds with parallel normalized mean curvature vector field (pnmc submanifolds) in the product spaces Sn×R, where Sn is the unit Euclidean sphere, is proved. As an application, an integral inequality for pnmc closed submanifolds in Sn×R is obtained. By this, it is shown that the sharpness in this inequality is attained, in the totally umbilical hypersurfaces, and in a certain family of standard tori of the form S1(1−r2)×Sn−1(r) with 0<r<1.

Three-Complex Numbers and Related Algebraic Structures

Three-complex numbers are introduced for using a geometric vector product in the three-dimensional Euclidean vector space R3 and proving its equivalence with a spherical coordinate product. Based upon the definitions of the geometric power and geometric exponential functions, some Euler-type trigonometric representations of three-complex numbers are derived. Further, a general l23−complex algebraic structure together with its matrix, polynomial and variable basis vector representations are considered. Then, the classes of lp3-complex numbers are introduced. As an application, Euler-type formulas are used to construct directional probability laws on the Euclidean unit sphere in R3.

Ограниченный оператор сдвига для (k, 1)-обобщенного преобразования Фурье

В пространствах с весом Данкля $v_k(x)$ степенного типа на $\mathbb{R}^d$, определяемым системой корней и неотрицательной функцией кратности $k$, инвариантной относительно конечной группы отражений, построен содержательный гармонический анализ. Классический анализ Фурье на евклидовом пространстве соответствует случаю $k\equiv 0$. В 2012 году Салем Бен Саид, Кобаяши и Орстед определили двупараметрическое $(k,a)$-обобщенное преобразование Фурье, действующее в пространствах с весом $|x|^{a-2}v_k(x)$, $a&gt;0$. Наиболее интересны случаи $a=2$ и $a=1$. При $a=2$ обобщенное преобразование Фурье совпадает с преобразованием Данкля и оно хорошо изучено. В случае $a=1$гармонический анализ, важный, в частности, в задачах квантовой механики, изучен пока еще не достаточно. Одним из существенных элементов гармонического анализа является ограниченный оператор сдвига, позволяющий определить свертку и структурные характеристики функций. При $a=1$ имеется оператор сдвига $\tau^yf(x)$. Его $L^p$-ограниченность недавно установлена Салемом Бен Саидом и Делеавалом, но только на радиальных функциях и при $1\le p\le 2$. В настоящей работе предложен новый оператор обобщенного сдвига $T^tf(x)$. Он получается интегрированием оператора $\tau^yf(x)$ по единичной евклидовой сфере по переменной $y'$, $|y'|=1$, $y=ty'$. Мы доказываем, что он положителен на функциях из пространства Шварца $\mathcal{S}(\mathbb{R}^d)$, для него $T^t1=1$ и он допускает представление с вероятностной мерой. Отсюда мы выводим его $L^p$-ограниченность для всех $1\le p&lt;\infty$ и ограниченность на пространстве $C_b(\mathbb{R}^d)$ непрерывных ограниченных функций.

Spectrum of the Laplacian and the Jacobi operator on rotational CMC hypersurfaces of spheres

Let $M\subset \mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be a compact cmc rotational hypersurface of the $(n+1)$-dimensional Euclidean unit sphere. Denote by $|A|^2$ the square of the norm of the second fundamental form and $J(f)=-\Delta f-nf-|A|^2f$ the stability or Jacobi operator. In this paper we compute the spectra of their Laplace and Jacobi operators in terms of eigenvalues of second order Hill's equations. For the minimal rotational examples, we prove that the stability index --the numbers of negative eigenvalues of the Jacobi operator counted with multiplicity -- is greater than $3 n+4$ and we also prove that there are at least 2 positive eigenvalues of the Laplacian of $M$ smaller than $n$. When $H$ is not zero, we have that every non-flat CMC rotational immersion is generated by rotating a planar profile curve along a geodesic called the axis of rotation. Let $m$ be the number of points where the maximal distance from this profile curve to the origin is achieved (we assume that the coordinates of the plane containing the profile curve has been set up so that the axis of rotation goes through the origin). Let $l$ be be the wrapping number of the profile curve. We show that the number of negative eigenvalues of the operator $J$ counted with multiplicity is at least $(2l-1)n+(2m-1)$. This result was proven for the case $n=2$ by Rossman and Sultana. They called $m$ the number of bulges or the number of necks. We will slightly change the definition of $l$ to include immersed examples that contain the axis of rotation.

Continuous valuations on the space of Lipschitz functions on the sphere

We study real-valued valuations on the space of Lipschitz functions over the Euclidean unit sphere Sn−1. After introducing an appropriate notion of convergence, we show that continuous valuations are bounded on sets which are bounded with respect to the Lipschitz norm. This fact, in combination with measure theoretical arguments, will yield an integral representation for continuous and rotation invariant valuations on the space of Lipschitz functions over the 1-dimensional sphere.