Euclidean Unit Ball Research Articles

Strengthened inequalities for the mean width and the ℓ‐norm

Barthe proved that the regular simplex maximizes the mean width of convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball; or equivalently, the regular simplex maximizes the ℓ-norm of convex bodies whose Löwner ellipsoid (minimal volume ellipsoid containing the body) is the Euclidean unit ball. Schmuckenschläger verified the reverse statement; namely, the regular simplex minimizes the mean width of convex bodies whose Löwner ellipsoid is the Euclidean unit ball. Here we prove stronger stability versions of these results. We also consider related stability results for the mean width and the ℓ-norm of the convex hull of the support of centered isotropic measures on the unit sphere.

The Littlewood-Offord problem for Markov chains

The celebrated Littlewood-Offord problem asks for an upper bound on the probability that the random variable ε1v1+⋯+εnvn lies in the Euclidean unit ball, where ε1,…,εn∈{−1,1} are independent Rademacher random variables and v1,…,vn∈Rd are fixed vectors of at least unit length. We extend some known results to the case that the εi are obtained from a Markov chain, including the general bounds first shown by Erdős in the scalar case and Kleitman in the vector case, and also under the restriction that the vi are distinct integers due to Sárközy and Szemeredi. In all extensions, the upper bound includes an extra factor depending on the spectral gap and additional dependency on the dimension. We also construct a pseudorandom generator for the Littlewood-Offord problem using similar techniques.

Best Low-rank Approximations and Kolmogorov $n$-widths

We relate the problem of best low-rank approximation in the spectral norm for a matrix $A$ to Kolmogorov $n$-widths and corresponding optimal spaces. We characterize all the optimal spaces for the image of the Euclidean unit ball under $A$ and we show that any orthonormal basis in an $n$-dimensional optimal space generates a best rank-$n$ approximation to $A$. We also present a simple and explicit construction to obtain a sequence of optimal $n$-dimensional spaces once an initial optimal space is known. This results in a variety of solutions to the best low-rank approximation problem and provides alternatives to the truncated singular value decomposition. This variety can be exploited to obtain best low-rank approximations with problem-oriented properties.

TWO NOTABLE CLASSES OF PROJECTIVE VECTOR FIELDS

Here, we find some necessary conditions for a projective vector field on a Randers metric to preserve the non-Riemannian quantities $\Xi$ and $H$.They are known in the contexts as the $C$-projective and $H$-projective vector fields. We find all projective vector fields of the Funk type metrics on the Euclidean unit ball $\mathbb{B}^n(1)$.

Orthogonal polynomial projection error in Dunkl–Sobolev norms in the ball

We study approximation properties of weighted L2-orthogonal projectors onto spaces of polynomials of bounded degree in the Euclidean unit ball, where the weight is of the reflection-invariant form (1−‖x‖2)α∏i=1d|xi|γi, α,γ1,…,γd>−1. Said properties are measured in Dunkl–Sobolev-type norms in which the same weighted L2 norm is used to control all the involved differential–difference Dunkl operators, such as those appearing in the Sturm–Liouville characterization of similarly weighted L2-orthogonal polynomials, as opposed to the partial derivatives of Sobolev-type norms.

The radii of sections of origin-symmetric convex bodies and their applications

Let V and W be any convex and origin-symmetric bodies in Rn . Assume that for some A ∈LRn→Rn, detA≠0, V is contained in the ellipsoid A−1B2n, where B2n is the unit Euclidean ball. We give a lower bound for the W-radius of sections of A−1V in terms of the spectral radius of A∗A and the expectations of ‖⋅‖V and ‖⋅‖Wo with respect to Haar measure on Sn−1⊂Rn. It is shown that the respective expectations are bounded as n→∞ in many important cases. As an application we offer a new method of evaluation of n-widths of multiplier operators. As an example we establish sharp orders of n-widths of multiplier operators Λ:LpMd→LqMd, 1<q≤2≤p<∞ on compact homogeneous Riemannian manifolds Md. Also, we apply these results to prove the existence of flat polynomials on Md.

Theorems of Carathéodory, Helly, and Tverberg Without Dimension

We initiate the study of no-dimensional versions of classical theorems in convexity. One example is Carathéodory’s theorem without dimension: given an n-element set P in a Euclidean space, a point $$a \in {{\,\mathrm{{\texttt {conv}}}\,}}P$$ , and an integer $$r \le n$$ , there is a subset $$Q\subset P$$ of r elements such that the distance between a and $${{\,\mathrm{{\texttt {conv}}}\,}}Q$$ is less than $${{\,\mathrm{{\texttt {diam}}}\,}}P/\sqrt{2r}$$ . In an analoguos no-dimension Helly theorem a finite family $$\mathcal {F}$$ of convex bodies is given, all of them are contained in the Euclidean unit ball of $$\mathbb {R}^d$$ . If $$k\le d$$ , $$|\mathcal {F}|\ge k$$ , and every k-element subfamily of $$\mathcal {F}$$ is intersecting, then there is a point $$q \in \mathbb {R}^d$$ which is closer than $$1/\sqrt{k}$$ to every set in $$\mathcal {F}$$ . This result has several colourful and fractional consequences. Similar versions of Tverberg’s theorem and some of their extensions are also established.

Loewner chains and nonlinear resolvents of the Carathéodory family on the unit ball in [formula omitted

In this paper we study various properties of nonlinear resolvents of holomorphic mappings in the Carathéodory family M(Bn), where Bn is the Euclidean unit ball in Cn. First, we prove certain characterizations of inverse Loewner chains f(z,t)=e−∫0ta(τ)dτz+⋯ on Bn×[0,∞), where a:[0,∞)→C is a locally Lebesgue integrable function such that ℜa(t)>0 for a.e. t≥0, and which satisfies a natural assumption. Next, we prove that if f∈M(Bn), then the nonlinear resolvent family {Jr}r≥0 is an inverse Loewner chain on Bn and the associated Herglotz vector field is of divergence type, where Jr=Jr[f]=(In+rf)−1, r≥0. This result is a generalization to higher dimensions of a recent result due to Elin, Shoikhet and Sugawa. We prove that (1+r)Jr can be embedded as the first element of a normal Loewner chain and the family {(1+r)Jr[f]:0≤r<∞,f∈M(Bn)} is a compact subset of S0(Bn). Also, we deduce that the shearing of (1+r)Jr (r≥0) associated with the family M(B2) is quasi-convex of type A and also starlike of order 4/5. We give a sufficient condition for (1+r)Jr to be quasiconformal on Bn and to be extended to a quasiconformal homeomorphism of Cn onto itself. Finally, sharp coefficient bounds for the nonlinear resolvent families of certain subsets of M(Bn), and also examples of support points of the compact families generated by nonlinear resolvent mappings on B2, will be obtained.

A boundary Schwarz lemma for mappings from the unit polydisc to irreducible bounded symmetric domains

AbstractIn this paper, we obtain a boundary Schwarz lemma for C1 (pluriharmonic, holomorphic) mappings from the unit polydisc in to irreducible bounded symmetric domains realized as the unit ball of an N‐dimensional simple JB*‐triple X. In particular, we obtain a version of the boundary Schwarz lemma for C1 (pluriharmonic, holomorphic) mappings from into the Euclidean unit ball in . These results are generalizations of recent results regarding boundary Schwarz lemma in higher dimensions.

A Schwarz lemma at the boundary on complex Hilbert balls and applications to starlike mappings

In this paper, we prove a Schwarz lemma at the boundary for holomorphic mappings f between Hilbert balls, and obtain related consequences. Especially, we obtain estimations of ∥Df(z0)∥ on the holomorphic tangent space for holomorphic mappings f or for homogeneous polynomial mappings f between Hilbert balls. Next, we prove the boundary rigidity theorem for holomorphic self-mappings of a Hilbert ball which have an interior fixed point. We obtain two distortion theorems for various subclasses of starlike mappings on the Euclidean unit ball $${\mathbb{B}^n}$$ in ℂn, as applications of the boundary Schwarz lemma for holomorphic mappings between the Euclidean unit balls. Finally, certain coefficient bounds for subclasses of starlike mappings on the unit ball of a complex Hilbert space are also obtained as an application of the estimation of ∥Df(z0)∥ on the holomorphic tangent space for homogeneous polynomial mappings f between Hilbert balls.

Support points for families of univalent mappings on bounded symmetric domains

In this paper we study some extremal problems for the family $S_g^0(\mathbb{B}_X)$ of normalized univalent mappings with $g$-parametric representation on the unit ball $\mathbb{B}_X$ of an $n$-dimensional JB$^*$-triple $X$ with $r\geq 2$, where $r$ is the rank of $X$ and $g$ is a convex (univalent) function on the unit disc $\mathbb{U}$, which satisfies some natural assumptions. We obtain sharp coefficient bounds for the family $S_g^0(\mathbb{B}_X)$, and examples of bounded support points for various subsets of $S_g^0(\mathbb{B}_X)$. Our results are generalizations to bounded symmetric domains of known recent results related to support points for families of univalent mappings on the Euclidean unit ball $\mathbb{B}^n$ and the unit polydisc $\mathbb{U}^n$ in $\mathbb{C}^n$. Certain questions will be also mentioned. Finally, we point out sharp coefficient bounds and bounded support points for the family $S_g^0(\mathbb{B}^n)$ and for special compact subsets of $S_g^0(\mathbb{B}^n)$, in the case $n\geq 2$.

A note on norms of signed sums of vectors

Abstract Improving a result of Hajela, we show for every function f with lim n→∞ f(n) = ∞ and f(n) = o(n) that there exists n 0 = n 0(f) such that for every n ⩾ n 0 and any S ⊆ {–1, 1} n with cardinality |S| ⩽ 2 n/f(n) one can find orthonormal vectors x 1, …, xn ∈ ℝ n satisfying ∥ ε 1 x 1 + ⋯ + ε n x n ∥ ∞ ⩾ c log ⁡ f ( n ) $\begin{array}{} \displaystyle \|\varepsilon_1x_1+\dots+\varepsilon_nx_n\|_{\infty }\geqslant c\sqrt{\log f(n)} \end{array}$ for all (ε 1, …, εn ) ∈ S. We obtain analogous results in the case where x 1, …, xn are independent random points uniformly distributed in the Euclidean unit ball B 2 n $\begin{array}{} \displaystyle B_2^n \end{array}$ or in any symmetric convex body, and the ℓ ∞ n $\begin{array}{} \displaystyle \ell_{\infty }^n \end{array}$ -norm is replaced by an arbitrary norm on ℝ n .

The Gromov-Hausdorff hyperspace of a Euclidean space

We investigate the hyperspace GH(Rn) of the isometry classes of all non-empty compact subsets of a Euclidean space in the Gromov-Hausdorff metric. It is proved that for any n≥1, GH(Rn) is homeomorphic to the orbit space 2Rn/E(n) of the hyperspace 2Rn of all non-empty compact subsets of a Euclidean space Rn equipped with the Hausdorff metric and the natural action of the Euclidean group E(n). This is further applied to prove that 2Rn/E(n) is homeomorphic to the open cone OCone(Ch(Bn)/O(n)), where Ch(Bn) stands for the set of all A∈2Rn for which the closed Euclidean unit ball Bn is the least circumscribed ball (the Chebyshev ball). These results lead to determine the complete topological structure of GH(Rn) for n≤2, namely, we prove that GH(Rn) is homeomorphic to the Hilbert cube with a removed point. We also prove that for n≤2, GH(Bn) is homeomorphic to the Hilbert cube.

Intrinsic and Dual Volume Deviations of Convex Bodies and Polytopes

AbstractWe establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced by the Wills functional and apply it to derive asymptotically sharp bounds for approximating the ball in high dimensions. Remarkably, it turns out that there is a polytope that is almost optimal with respect to all intrinsic volumes simultaneously, up to absolute constants.Finally, we establish asymptotic formulas for the best approximation of smooth convex bodies by polytopes with respect to a distance induced by dual volumes, which originate from Lutwak’s dual Brunn–Minkowski theory.

On Some Problems for a Simplex and a Ball in $${{\mathbb{R}}^{n}}$$

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 the homothety with center of homothety in the center of gravity of $$S$$ and ratio of homothety $$\tau $$. 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 $$\xi (C;S) = (n + 1)\mathop {\max }\limits_{1 \leqslant j \leqslant n + 1} \mathop {\max }\limits_{x \in C} ( - {{\lambda }_{j}}(x)) + 1$$ (if $$C { \text{⊄} }S$$), $$\alpha (C;S) = \sum\nolimits_{j = 1}^{n + 1} {\mathop {\max }\limits_{x \in C} ( - {{\lambda }_{j}}(x)) + 1.} $$ Here $${{\lambda }_{j}}$$ are linear functions 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 has 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 $${{B}_{n}} = \{ x:\left| {\left| x \right|} \right| \leqslant 1\} ,$$ where $$\left| {\left| x \right|} \right| = \mathop {\left( {\sum\nolimits_{i = 1}^n {x_{i}^{2}} \,} \right)}\nolimits^{1/2} .$$ 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 $$\alpha ({{B}_{n}};S) = \sum\nolimits_{j = 1}^{n + 1} {{{{\left( {\sum\nolimits_{i = 1}^n {l_{{ij}}^{2}} } \right)}}^{{{\text{1}}{\text{/}}{\text{2}}}}}} $$. The minimal possible value of each characteristic $$\xi ({{B}_{n}};S)$$ 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 $$C = {{Q}_{n}}$$.

Geometric Estimates in Interpolation on an n-Dimensional Ball

Suppose \(n\in {\mathbb N}\). Let \(B_n\) be a Euclidean unit ball in \({\mathbb R}^n\) given by the inequality \(\|x\|\leq 1\), \(\|x\|:=\left(\sum\limits_{i=1}^n x_i^2\right)^{\frac{1}{2}}\). By \(C(B_n)\) we mean a set of continuous functions \(f:B_n\to{\mathbb R}\) with the norm \(\|f\|_{C(B_n)}:=\max\limits_{x\in B_n}|f(x)|\). The symbol \(\Pi_1\left({\mathbb R}^n\right)\) denotes a set of polynomials in \(n\) variables of degree \(\leq 1\), i.e. linear functions upon \({\mathbb R}^n\). Assume that \(x^{(1)}, \ldots, x^{(n+1)}\) are vertices of an \(n\)-dimensional nondegenerate simplex \(S\subset B_n\). The interpolation projector \(P:C(B_n)\to \Pi_1({\mathbb R}^n)\) corresponding to \(S\) is defined by the equalities \(Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).\) Denote by \(\|P\|_{B_n}\) the norm of \(P\) as an operator from \(C(B_n)\) on to \(C(B_n)\). Let us define \(\theta_n(B_n)\) as the minimal value of \(\|P\|_{B_n}\) under the condition \(x^{(j)}\in B_n\). We describe the approach in which the norm of the projector can be estimated from the bottom through the volume of the simplex. Let \(\chi_n(t):=\frac{1}{2^nn!}\left[ (t^2-1)^n \right] ^{(n)}\) be the standardized Legendre polynomial of degree \(n\). We prove that \(\|P\|_{B_n}\geq\chi_n^{-1}\left(\frac{vol(B_n)}{vol(S)}\right).\) From this, we obtain the equivalence \(\theta_n(B_n)\) \(\asymp\) \(\sqrt{n}\). Also we estimate the constants from such inequalities and give the comparison with the similar relations for linear interpolation upon the \(n\)-dimensional unit cube. These results have applications in polynomial interpolation and computational geometry.

Euler partial differential equations and Schwartz distributions

Euler operators are partial differential operators of the formP(θ)P(\theta )wherePPis a polynomial andθj=xj∂/∂xj\theta _j = x_j \partial /\partial x_j. They are surjective on the space of temperate distributions onRd\mathbb {R}^d. We show that this is, in general, not true for the space of Schwartz distributions onRd\mathbb {R}^d,d≥3d\ge 3, ford=1d=1; however, it is true. It is also true for the space of distributions of finite order onRd\mathbb {R}^dand on certain open setsΩ⊂Rd\Omega \subset \mathbb {R}^d, like the Euclidean unit ball.

Homogeneous Hermitian holomorphic vector bundles and the Cowen-Douglas class over bounded symmetric domains

It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic Lie algebra on finite dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. Our first main result is the construction of an explicit differential operator intertwining the bundle with the direct sum of its factors. Next, we study Hilbert spaces of sections of these bundles. We use this to get, in particular, a full description and a similarity theorem for homogeneous n-tuples of operators in the Cowen-Douglas class of the Euclidean unit ball in Cn.

Brosamler’s formula revisited and extensions

Brosamler’s formula gives a probabilistic representation of the solution of the Neumann problem for the Laplacian on a smooth bounded domain $$D\subset \mathbb {R}^n$$ in terms of the reflecting Brownian motion in D. The original proof, as well as other proofs in the literature (e.g., in the case of Lipschitz domains), are based on potential theory (transition densities of the reflecting Brownian motion). We give new proofs of Brosamler’s formula using (path trajectories of) stochastic processes. More precisely, we use a connection between the Dirichlet and the Neumann boundary problems, and the explicit description of the reflecting Brownian motion and its boundary local time in terms of the free Brownian motion. The results are obtained in the case of the Euclidean unit ball in any dimension and in the case of smooth $$C^{1,\alpha }$$ planar simply connected domains, for continuous boundary data, and then extended to the case of bounded measurable data, respectively integrable boundary data. A new Brosamler-type formula in terms of the free Brownian motion is also given.

Vanishing theorems for the cohomology groups of free boundary submanifolds

In this paper, we prove that there exists a universal constant C, depending only on positive integers $$n\ge 3$$ and $$p\le n-1$$ , such that if $$M^n$$ is a compact free boundary submanifold of dimension n immersed in the Euclidean unit ball $$\mathbb {B}^{n+k}$$ whose size of the traceless second fundamental form is less than C, then the pth cohomology group of $$M^n$$ vanishes. Also, employing a different technique, we obtain a rigidity result for compact free boundary surfaces minimally immersed in the unit ball $$\mathbb {B}^{2+k}$$ .