Euclidean Unit Ball Research Articles

Equivariant insight into hyperspaces of convex bodies

Let cc(Rn) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space Rn endowed with the Hausdorff metric, and let cc(Bn)={A∈cc(Rn)|A⊂Bn}, where Bn is the closed unit ball of Rn. Let cb(Rn) be the subset of cc(Rn) consisting of all convex bodies, i.e., of sets A∈cc(Rn) with non-empty interior. In this survey article we reveal fundamental properties of the natural action Aff(n)↷cb(Rn) of the affine group that leads to structure results for cb(Rn) and for several important subsets of it. Orbit spaces of cb(Rn) and its subsets by some appropriate subgroups G<Aff(n) are homeomorphic to such important objects like the Hilbert cube or the Banach-Mazur compacta. Among other hyperspaces studied here are cw(Rn) - the hyperspace of convex bodies of constant width, Bp - the hyperspace of convex bodies with smooth boundary of positive Gaussian curvature, L(n) - the hyperspace of convex bodies for which the Euclidean unit ball is the Löwner ellipsoid, cˇ(n) - the hyperspace of convex sets for which the Euclidean unit ball is the Chebyshev ball, etc. The paper contains also several new results concerning equivariant extension properties of the hyperspaces in question, as well as the homeomorphism type of the hyperspace cb(Bn)=cb(Rn)∩cc(Bn) and its orbit spaces. Equivariant extension properties of these hyperspaces are applied to provide a very short proof of the classical Grünbaum Conjecture from Convex Geometry. Along the paper seventeen open problems are raised.

On the existence of symplectic barriers

In this note we establish the existence of a new type of rigidity of symplectic embeddings coming from obligatory intersections with symplectic planes. In particular, we prove that if a Euclidean ball is symplectically embedded in the Euclidean unit ball, then it must intersect a sufficiently fine grid of two-codimensional pairwise disjoint symplectic planes. Inspired by analogous terminology for Lagrangian submanifolds, we refer to these obstructions as symplectic barriers.

Regular ellipsoids and a Blaschke-Santaló-type inequality for projections of non-symmetric convex bodies

It is shown that every not-necessarily symmetric convex body K in Rn has an affine image K˜ of K such that the covering numbers of K˜ by growing dilates of the unit Euclidean ball, as well as those of the unit Euclidean ball by growing dilates of K˜, decrease in a regular way. This extends to the non-symmetric case a famous theorem by Pisier, albeit with worse estimates on the rate of decrease of the covering numbers. The affine image K˜ can be chosen to have either barycentre or Santaló point at the origin.In the proof we use Pisier's theorem as a black box, as well as a suggested approach by Klartag and V. Milman. A key new ingredient is Blaschke-Santaló-type inequalities for projections of a body K˜ with Santaló point at the origin, which could be of independent interest. Unlike the application to covering, these (as well as the analogous inequalities for centred convex bodies that were already considered by Klartag and Milman [28]) can be shown to be optimal up to absolute constants.We also present an application to results around the mean norm of isotropic (not-necessarily symmetric) convex bodies.

An Algorithm Based on Compute Unified Device Architecture for Estimating Covering Functionals of Convex Bodies

In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a longstanding open problem from Convex and Discrete Geometry, it is essential to estimate covering functionals of convex bodies effectively. Recently, He et al. and Yu et al. provided two deterministic global optimization algorithms having high computational complexity for this purpose. Since satisfactory estimations of covering functionals will be sufficient in Zong’s program, we propose a stochastic global optimization algorithm based on CUDA and provide an error estimation for the algorithm. The accuracy of our algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean unit disc, the three-dimensional Euclidean unit ball, the regular tetrahedron, and the regular octahedron. We also present estimations of covering functionals for the regular dodecahedron and the regular icosahedron.

Quantitative Steinitz theorem: A polynomial bound

AbstractThe classical Steinitz theorem states that if the origin belongs to the interior of the convex hull of a set , then there are at most points of whose convex hull contains the origin in the interior. Bárány, Katchalski, and Pach proved the following quantitative version of Steinitz's theorem. Let be a convex polytope in containing the standard Euclidean unit ball . Then there exist at most vertices of whose convex hull satisfies with . They conjectured that holds with a universal constant . We prove , the first polynomial lower bound on . Furthermore, we show that is not greater than .

Generalized parametric representation and extension operators

We consider conditions under which mappings of the open Euclidean unit ball of Cn with generalized parametric representation with respect to an appropriately defined locally integrable function A:[0,∞)→L(Cn), where L(Cn) is the algebra of linear operators from Cn into Cn, on an interval [s,∞), s≥0, are extended to mappings of the same type defined in a higher-dimensional ball (with respect to a related operator-valued function) using two classes of extension operators, namely perturbations of the Pfaltzgraff–Suffridge extension operator and modifications of the Roper–Suffridge extension operator introduced by the author. Extensions of Loewner chains of order p∈[1,∞], i.e., non-normalized Loewner chains that satisfy a locally uniform local Lp-continuity condition on their real parameter, using these operators have previously been considered by the author, but, unlike in the case of standard parametric representation, mappings with generalized parametric representation may not be as nicely compatible with Loewner chains. We will consider the ramifications of this. As part of our work, we will see that A-normalized Loewner chains and evolution families will have order p when ‖A(⋅)‖ is locally Lp. In addition, we will completely characterize generalized parametric representation in the open unit disk. This characterization involves families of bounded univalent functions and allows us to show that the modified Roper–Suffridge extension operators extend families of univalent functions bounded by some M>1 to biholomorphic mappings bounded by M under certain conditions as a consequence of our main extension results.

Extension operators and support points associated with g-Loewner chains on complex Banach spaces

We prove that the Roper-Suffridge extension operator preserves g-parametric representation on general domains Bp1,p2={(z1,w)∈C×X:|z1|p1+‖w‖Xp2<1}, where X is a complex Banach space and p1,p2≥1. This result generalizes recent result from p1=2 to 1≤p1<∞. Next, we obtain a rigidity property of the Roper-Suffridge extension operator for the preservation of convexity. Finally, some sharp coefficient bounds for univalent mappings with g-parametric representation on the unit p-ball of Cn (1<p≤∞) are also obtained. As applications, we establish bounded support points with g-parametric representation on the unit p-ball. These results cover recent results related to bounded support points for families of biholomorphic mappings on the Euclidean unit ball for p=2, and on the unit polydisc for p=∞, respectively.

An intrinsic volume metric for the class of convex bodies in ℝn

A new intrinsic volume metric is introduced for the class of convex bodies in [Formula: see text]. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes with a restricted number of vertices under this metric. This result improves the best known estimate, and shows that dropping the restriction that the polytope is contained in the ball or vice versa improves the estimate by at least a factor of dimension. The same phenomenon has already been observed in the special cases of volume, surface area and mean width approximation of the ball.

Linear isoperimetric inequalities for free boundary submanifolds in a geodesic ball

We derive linear isoperimetric inequalities for free boundary submanifolds in a geodesic ball of a Riemannian manifold in terms of the modified volume. It is known that the twice of the area of a free boundary minimal surface in a Euclidean unit ball is equal to the length of its boundary. This can be extended to space forms by using our linear isoperimetric inequalities for the modified volume. Moreover, we obtain a sharp lower bound for the modified volume of free boundary minimal surfaces in a geodesic ball of the upper hemisphere mathbb {S}_+^3. Finally, it is proved that the monotonicity property still holds for the modified volume of any submanifold in a complete simply connected Riemannian manifold with sectional curvature bounded above by a constant.

On an Application of Jack’s Lemma, Starlikeness and k-Fold Symmetry in {mathbb {C}}^{n}

It is known that the starlikeness plays a central role in complex analysis, similarly as the convexity in functional analysis. However, if we consider the biholomorphisms between domains in {mathbb {C}}^{n}, apart from starlikeness of domains, various symmetries are also important. This follows from the Poincaré theorem showing that the Euclidean unit ball is not biholomorphically equivalent to a polydisc in {mathbb {C}}^{n},n>1. From this reason the second author in 2003 considered some families of locally biholomorphic mappings defined in the Euclidean open unit ball using starlikeness factorization and a notion of k-fold symmetry. The 2017 paper of both authors contains some results on the absorption by a family S(k),kge 2, of the above kind, the families of mappings biholomorphic starlike (convex) and vice versa. In the present paper there is given a new sufficient criterion for a locally biholomorphic mapping f, from the Euclidean ball {mathbb {B}}^{n} into {mathbb {C}}^{n}, to belong to the family S(k),kge 2. The result is obtained using an n-dimensional version of Jack’s Lemma.

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.

A necessary condition for prescribing mean curvature equations in 𝔹ⁿ

Let ( B n , g 0 ) (\mathbb {B}^n,g_0) be the n n -dimensional Euclidean unit ball with n ⩾ 3 n\geqslant 3 . We obtain a necessary condition on the existence of conformal metric in B n \mathbb {B}^n with zero scalar curvature and prescribed mean curvature function on the boundary.

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.

A rigidity theorem at the boundary for holomorphic mappings with values in finite dimensional bounded symmetric domains

AbstractLet be a bounded symmetric domain realized as the open unit ball of a finite dimensional JB*‐triple X. In this paper, we obtain a rigidity theorem at the boundary for holomorphic mappings from a balanced domain G in a complex Banach space E into . We also obtain a rigidity theorem at the boundary for holomorphic self‐mappings of . Our results give generalizations of the recent results obtained on the Euclidean unit ball or the unit polydisc in .

Some sharp Schwarz-Pick type estimates and their applications of harmonic and pluriharmonic functions

The purpose of this paper is to study the Schwarz-Pick type inequalities for harmonic or pluriharmonic functions. By analogy with the generalized Khavinson conjecture, we first give some sharp estimates of the norm of harmonic functions from the Euclidean unit ball in Rn into the unit ball of the real Minkowski space. Next, we give several sharp Schwarz-Pick type inequalities for pluriharmonic functions from the Euclidean unit ball in Cn or from the unit polydisc in Cn into the unit ball of the Minkowski space. Furthermore, we establish some sharp coefficient type Schwarz-Pick inequalities for pluriharmonic functions defined in the Minkowski space. Finally, we use the obtained Schwarz-Pick type inequalities to discuss the Lipschitz continuity, the Schwarz-Pick type lemmas of arbitrary order and the Bohr phenomenon of harmonic or pluriharmonic functions.

Reverse Alexandrov–Fenchel inequalities for zonoids

The Alexandrov–Fenchel inequality bounds from below the square of the mixed volume [Formula: see text] of convex bodies [Formula: see text] in [Formula: see text] by the product of the mixed volumes [Formula: see text] and [Formula: see text]. As a consequence, for integers [Formula: see text] with [Formula: see text] the product [Formula: see text] of suitable powers of the volumes [Formula: see text] of the convex bodies [Formula: see text], [Formula: see text], is a lower bound for the mixed volume [Formula: see text], where [Formula: see text] is the multiplicity with which [Formula: see text] appears in the mixed volume. It has been conjectured by Betke and Weil that there is a reverse inequality, that is, a sharp upper bound for the mixed volume [Formula: see text] in terms of the product of the intrinsic volumes [Formula: see text], for [Formula: see text]. The case where [Formula: see text], [Formula: see text], [Formula: see text] has recently been settled by the present authors (2020). The case where [Formula: see text], [Formula: see text], [Formula: see text] has been treated by Artstein-Avidan et al. under the assumption that [Formula: see text] is a zonoid and [Formula: see text] is the Euclidean unit ball. The case where [Formula: see text], [Formula: see text] is the unit ball and [Formula: see text] are zonoids has been considered by Hug and Schneider. Here, we substantially generalize these previous contributions, in cases where most of the bodies are zonoids, and thus we provide further evidence supporting the conjectured reverse Alexandrov–Fenchel inequality. The equality cases in all considered inequalities are characterized. More generally, stronger stability results are established as well.

On transfinite diameters in $\mathbb{C}^{d}$ for generalized notions of degree

We give a general formula for the $C$-transfinite diameter $\delta_C(K)$ of a compact set $K\subset \mathbb{C}^2$ which is a product of univariate compacta where $C\subset (\mathbb{R}^+)^2$ is a convex body. Along the way we prove a Rumely type formula relating $\delta_C(K)$ and the $C$-Robin function $\rho_{V_{C,K}}$ of the $C$-extremal plurisubharmonic function $V_{C,K}$ for $C \subset (\mathbb{R}^+)^2$ a triangle $T_{a,b}$ with vertices $(0,0)$, $(b,0)$, $(0,a)$. Finally, we show how the definition of $\delta_C(K)$ can be extended to include many nonconvex bodies $C\subset \mathbb{R}^d$ for $d$-circled sets $K\subset \mathbb{C}^d$, and we prove an integral formula for $\delta_C(K)$ which we use to compute a formula for $\delta_C(\mathbb{B})$ where $\mathbb{B}$ is the Euclidean unit ball in $\mathbb{C}^2$.

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.

Uniqueness Results for Free Boundary Minimal Hypersurfaces in Conformally Euclidean Balls and Annular Domains

In this paper, we prove that a flat free boundary minimal n-disk, $$n\ge 3$$ , in the unit Euclidean ball $${\mathbb {B}}^{n+1}$$ is the unique compact free boundary minimal hypersurface whose squared norm of the second fundamental form is less than either $$\frac{n^2}{4}$$ or $$\frac{(n-2)^2}{4|x|^2}$$ . Moreover, we prove analogous results for compact free boundary minimal hypersurfaces in annular domains or balls with a conformally Euclidean metric.

Horosphere-invariant mappings of the unit ball in [formula omitted

Horospheres, particular ellipsoids internally tangent to the boundary of the Euclidean unit ball B of Cn, play a pivotal role in the Denjoy–Wolff theory of holomorphic self-mappings of B. We use horospheres to introduce a new family of transforms on the class of normalized locally biholomorphic mappings of B into Cn in a vein similar to the Koebe transforms with respect to automorphisms of B. Like the Koebe transforms, these horosphere transforms preserve the property of convexity of a mapping. After studying the transforms with regard to convex mappings in particular, we give a broad characterization of all normalized holomorphic mappings that are invariant under the horosphere transforms. We also show that the notion of horosphere invariance for locally biholomorphic mappings is preserved by several commonly studied extension operators. We then apply our technique to provide a similar, simplified characterization of Koebe-invariant mappings. With these characterizations in hand, we show that, despite the existence of many mappings satisfying either type of invariance separately, the only normalized holomorphic mappings that enjoy both types of invariance are those mappings of the formf(z)=(z11−z1+Q(zˆ)(1−z1)2,zˆ(1−z1)) for z=(z1,zˆ)∈B, where zˆ=(z2,…,zn) and Q:Cn−1→C is a homogeneous polynomial of degree 2. For certain Q, these mappings are extreme points of the closed convex hull of the family of convex mappings of B.