On approximation by projections of polytopes with few facets

We provide an affirmative answer to a problem posed by Barvinok and Veomett in [4], showing that in general an n-dimensional convex body cannot be approximated by a projection of a section of a simplex of subexponential dimension. Moreover, we prove that for all 1 ≤ n ≤ N there exists an n-dimensional convex body B such that for every n-dimensional convex body K obtained as a projection of a section of an N-dimensional simplex one has $$d(B,K) \geqslant c\sqrt {\frac{n}{{\ln \frac{{2N\ln (2N)}}{n}}}} $$ , where d(·, ·) denotes the Banach-Mazur distance and c is an absolute positive constant. The result is sharp up to a logarithmic factor.     

Minimal time function and viscosity solutions

Two theorems in Ref. 1 are generalized. It is proved that, ifV(A,Γ) is the set of points that can be steered to the origin along a solution of the control systemx′=Ax?c, ifc(t)∈Γ, Γ is a compact subset ofR ⁿ , 0∈ intrelco Γ, and if a rank condition holds, then the minimal time functionT(·) is a viscosity solution of the Bellman equation $$\max \{ \left\langle {DT(x),\gamma - Ax} \right\rangle :\gamma \varepsilon co\Gamma \} - 1 = 0,x\varepsilon V(A,\Gamma )\backslash \{ 0\} ,$$ and of the Hàjek equation $$1 - \max \{ \left\langle {DT(x),\exp [ - AT(x)]} \right\rangle :\gamma \varepsilon co\Gamma \} = 0,x\varepsilon V(A,\Gamma ).$$      

Finite projective spaces and intersecting hypergraphs

Let ? be a family ofk-subsets on ann-setX andc be a real number 0 <c<1. Suppose that anyt members of ? have a common element (t ≧ 2) and every element ofX is contained in at mostc|?| members of ?. One of the results in this paper is (Theorem 2.9): If $$c = {{(q^{t - 1} + ... + q + 1)} \mathord{\left/ {\vphantom {{(q^{t - 1} + ... + q + 1)} {(q^t + ... + q + 1)}}} \right. \kern-\nulldelimiterspace} {(q^t + ... + q + 1)}}$$ . whereq is a prime power andn is sufficiently large, (n >n (k, c)) then The corresponding lower bound is given by the following construction. LetY be a (q ^t + ... +q + 1)-subset ofX andH ₁,H ₂, ...,H _|Y| the hyperplanes of thet-dimensional projective space of orderq onY. Let ? consist of thosek-subsets which intersectY in a hyperplane, i.e., ?={F∈( _k ^X ): there exists ani, 1≦i≦|Y|, such thatY∩F=H _i }.     

Inequalities for convex bodies and polar reciprocal lattices inR n II: Application ofK-convexity

The paper is a supplement to [2]. LetL be a lattice andU ano-symmetric convex body inR ⁿ . The Minkowski functional? ⁿ ofU, the polar bodyU ⁰, the dual latticeL ^*, the covering radius μ(L, U), and the successive minima λ _i ,i=1, …,n, are defined in the usual way. Let $\mathcal{L}_n $ be the family of all lattices inR ⁿ . Given a convex bodyU, we define $$\begin{gathered} mh(U){\text{ }} = {\text{ }}\sup {\text{ }}\max \lambda _i (L,U)\lambda _{n - i + 1} (L^* ,U^0 ), \hfill \\ {\text{ }}L \in \mathcal{L}_n 1 \leqslant i \leqslant n \hfill \\ lh(U){\text{ }} = {\text{ }}\sup {\text{ }}\lambda _1 (L,U) \cdot \mu (L^* ,U^0 ), \hfill \\ {\text{ }}L \in \mathcal{L}_n \hfill \\ \end{gathered} $$ and kh(U) is defined as the smallest positive numbers for which, given arbitrary $L \in \mathcal{L}_n $ andx∈R ⁿ /(L+U), somey∈L ^* with ∥y∥ _U ⁰?sd(xy,Z) can be found. It is proved $$C_1 n \leqslant jh(U) \leqslant C_2 nK(R_U^n ) \leqslant C_3 n(1 + \log n),$$ , for j=k, l, m, whereC ₁,C ₂,C ₃ are some numerical constants andK(R _U ⁿ ) is theK-convexity constant of the normed space (R ⁿ , ∥∥U). This is an essential strengthening of the bounds obtained in [2]. The bounds for lh(U) are then applied to improve the results of Kannan and Lovász [5] estimating the lattice width of a convex bodyU by the number of lattice points inU.     

A constructive description of Hölder classes on closed Jordan curves

Let Γ be a closed, Jordan, rectifiable curve, whose are length is commensurable with its subtending chord, leta ε int Γ, and let R_n(a) be the set of rational functions of degree ≤n, having a pole perhaps only at the pointa. Let Λ^α(Γ), 0 < α < 1, be the Hölder class on Γ. One constructs a system of weights γ_n(z) > 0 on Γ such that f∈Λ^α(Γ) if and only if for any nonnegative integer n there exists a function R_n, R_n ε R_n(a) such that ¦f(z) ? R_n(z)¦ ≤ c_f·γ_n(z), z ε Γ. It is proved that the weights γ_n cannot be expressed simply in terms of ρ ₁ ⁺ /n^(z) and ρ ₁ ^- /n^(z), the distances to the level lines of the moduli of the conformal mappings of ext Γ and int Γ on \(\mathbb{C}\backslash \mathbb{D}\) .     

Extreme points, support points and the Loewner variation in several complex variables

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class S0(Bn) of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f (·, 0) of a Loewner chain f (z, t) = etz + ··· such that {e-tf (·, t)}t 0 is a normal family on Bn. We show that if f (·, 0) is an extreme point (respectively a support point) of S0(Bn), then e-tf (·, t) is an extreme point of S0(Bn) for t 0 (respectively a support point of S0(Bn) for t ∈[0, t0] and some t0 > 0). This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to S0(Bn) and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.     

Vertical versus horizontal Poincaré inequalities on the Heisenberg group

Let ? = 〈a, b|a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric d _W (·, ·) associated to the generating set {a, b, a ^?1, b ^?1}. Letting B _n = {x ∈ ?: d _W (x, e _?) ? n} denote the corresponding closed ball of radius n ∈ ?, and writing c = [a, b] = aba ^?1 b ^?1, we prove that if (X, ‖ · ‖X) is a Banach space whose modulus of uniform convexity has power type q ∈ [2,∞), then there exists K ∈ (0, ∞) such that every f: ? → X satisfies $$\sum\limits_{k = 1}^{{n^2}} {\sum\limits_{x \in {B_n}} {\frac{{\left\| {f(x{c^k}) - f(x)} \right\|_X^q}}{{{k^{1 + q/2}}}}} } \leqslant K\sum\limits_{x \in {B_{21n}}} {(\left\| {f(xa) - f(x)} \right\|_X^q + \left\| {f(xb) - f(x)} \right\|_X^q)} $$ . It follows that for every n ∈ ? the bi-Lipschitz distortion of every f: B _n → X is at least a constant multiple of (log n)^1/q , an asymptotically optimal estimate as n → ∞.     

Algorithms for the vector maximization problem

We consider a convex setB inR ⁿ described as the intersection of halfspacesa _i ^T x ≦b _i (i ∈ I) and a set of linear objective functionsf _j =c _j ^T x (j ∈ J). The index setsI andJ are allowed to be infinite in one of the algorithms. We give the definition of theefficient points ofB (also called functionally efficient or Pareto optimal points) and present the mathematical theory which is needed in the algorithms. In the last section of the paper, we present algorithms that solve the following problems:

1. To decide if a given point inB is efficient.
2. To find an efficient point inB.
3. To decide if a given efficient point is the only one that exists, and if not, find other ones.
4. The solutions of the above problems do not depend on the absolute magnitudes of thec _j. They only describe the relative importance of the different activitiesx _i. Therefore we also consider $$\begin{gathered} \max G^T x \hfill \\ x efficient \hfill \\ \end{gathered} $$ for some vectorG.

     

R/S统计量重对数律的精确收敛速度

Let{Xn;n≥1}be a sequence of i.i.d, random variables with finite variance,Q（n）be the related R/S statistics. It is proved that lim ε↓0 ε^2 ∑n=1 ^8 n log n/1 P{Q（n）≥ε√2n log log n}=2/1 EY^2,where Y=sup0≤t≤1B（t）-inf0≤t≤sB（t）,and B（t） is a Brownian bridge.     

Fractal relaxed Dirichlet problems

Given aself similar fractal K ? ? ⁿ of Hausdorff dimension α>n?2, andc ₁>0, we give an easy and explicit construction, using the self similarity properties ofK, of a sequence of closed sets? _h such that for every bounded open setΩ?? ⁿ and for everyf ∈ L²(Ω) the solutions to $$\left\{ \begin{gathered} - \Delta u_h = f in \Omega \backslash \varepsilon _h \hfill \\ u_h = 0 on \partial (\Omega \backslash \varepsilon _h ) \hfill \\ \end{gathered} \right.$$ converge to the solution of the relaxed Dirichlet boundary value problem $$\left\{ \begin{gathered} - \Delta u + uc_1 \mathcal{H}_{\left| K \right.}^\alpha = f in \Omega \hfill \\ u = 0 on \partial \Omega \hfill \\ \end{gathered} \right.$$ (H _∣ ^α denotes the restriction of the α-dimensional Hausdorff measure toK). The condition α>n?2 is strict.     

On the maximum of the minimum of a sum

LetB denote the closure of a bounded open set of points inE ⁿ with Jordan content |B|>0 and letc>0 be constant. Typical of the expressions considered is $$M(N,c) = \max _{\left\{ {x_j } \right\}} \min _{x \in B} \sum\limits_{j = 1}^N {\left| {x - x_j } \right|^{ - c} } ,x_j \in E^n$$ Together with its analogs and extensions, the problem forc has a long history, associated with the names of Fekete, Leja, Pólya, Szegö, Frostman and Carleson, to mention just a few. It involves the notions of generalized capacity, transfinite diameter, and equilibrium potential. Here we consider the casec≧n and its extensions, for which the prior history seems less comprehensive. Illustrative of the results obtained are the three equations $$\mathop {\lim }\limits_{N \to \infty } \frac{{M(N,n)}}{{N\log N}} = \frac{{\omega (n)}}{{\left| B \right|}},\mathop {\lim }\limits_{N \to \infty } \frac{{M(N,c)}}{{N^{{c \mathord{\left/ {\vphantom {c n}} \right. \kern-\nulldelimiterspace} n}} }} = \frac{{L(n,c)}}{{\left| B \right|^{{c \mathord{\left/ {\vphantom {c n}} \right. \kern-\nulldelimiterspace} n}} }},\mathop {\lim }\limits_{c \to \infty } L(n,c)^{{1 \mathord{\left/ {\vphantom {1 c}} \right. \kern-\nulldelimiterspace} c}} = \mathop {\lim }\limits_{N \to \infty } \frac{{\left( {\left| B \right|/N} \right)^{{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-\nulldelimiterspace} n}} }}{{\varrho (N)}}$$ In the firstc=n and ω (n) is the volume of the unit ball. In the secondc>n and existence of the limit is asserted, 0<L(n,c)<∞. In the third, ? (N) is the smallest value such thatN spheres of radius ? (N) can coverB. The results would be unchanged if we requiredx _j ∈B instead ofx _j ∈E ⁿ in the definition ofM(N, c).     

Cantor-Lebesgue theorem for double trignometric series

Let ∥·∥ be a norm in R² and let γ be the unit sphere induced by this norm. We call a segment joining points x,y ε R² rational if (x₁ ? y₁)/(x₂ ? y₂) or (x₂ ? y₂)/(x₁ ? y₁) is a rational number. Let γ be a convex curve containing no rational segments. Satisfaction of the condition $$T_\nu (x) = \sum\nolimits_{\parallel n\parallel = \nu } {c_n e^{2\pi i(n_1 x_1 + n_2 x_2 )} } \to 0(\nu \to \infty )$$ in measure on the set e? [- 1/2,1/2)×[- 1/2, 1/2) =T² of positive planar measure implies ∥T _v ∥L₄ (T²) → 0(v → ∞). if, however, γ contains a rational segment, then there exist a sequence of polynomials {T _v } and a set E ? T², ¦E¦ > 0, such that T _v (x) → 0(v → ∞) on E; however, ¦c_n¦ ? 0 for ∥n∥ → ∞.     

On the entire self-shrinking solutions to Lagrangian mean curvature flow

The authors prove that the logarithmic Monge?CAmpère flow with uniformly bound and convex initial data satisfies uniform decay estimates away from time t?=?0. Then applying the decay estimates, we conclude that every entire classical strictly convex solution of the equation $$ \det D^{2}u=\exp\left\{n\left(-u+\frac{1}{2} \sum_{i=1}^{n}x_{i} \frac{\partial u}{\partial x_{i}} \right)\right\}, $$ should be a quadratic polynomial if the inferior limit of the smallest eigenvalue of the function |x|² D ² u at infinity has an uniform positive lower bound larger than 2(1 ? 1/n). Using a similar method, we can prove that every classical convex or concave solution of the equation $$ \sum_{i=1}^{n} \arctan\lambda_{i}=-u+\frac{1}{2} \sum_{i=1}^{n}x_{i} \frac{\partial u}{\partial x_{i}} $$ must be a quadratic polynomial, where ?? _i are the eigenvalues of the Hessian D ² u.     

Uniqueness of horospheres and geodesic cylinders in hyperbolic space

A hyperbolic analogon to Hartman’s characterization of orthogonal sphere cylinders is proved: Let Mn ? Hⁿ⁺¹ be a noncompact closed hypersurface with sectional curvature K ≥ 0 which bounds a convex set. Assume further H_r ≡ c for one normalized mean curvature. Then Mⁿ is a horosphere or a geodesic cylinder if $r{\leq}\ {2\over 3}\ (n+1)$ . For $r >\ {2\over 3}\ (n+1)$ the same follows but only if c lies in a specified interval which however covers the case of a horosphere. The argumentation is based on results of S.B. Alexander and R.B. Currier on the infinity set of certain convex hypersurfaces, the comparison with interior spindle surfaces, first eigenvalue estimates for Voss operators and variational properties of relevant curvature expressions.     

Dissipativity of some class of uncertain systems

We consider the system $$ \dot x = A\left( \cdot \right)x + B\left( \cdot \right)u, u = S\left( \cdot \right)x, t \geqslant t_0 , $$ where A(·) ∈ ? ^n×n , B(·) ? ^n×p , and S(·) ∈ ? ^p×n . The entries of matrices A(·), B(·), and S(·) are arbitrary bounded functionals. We consider the problem of constructing a matrix H > 0 and finding relations between the entries of the matrices B(·) and S(·) such that for a given constant matrix R the inequality $$ V\left( {x\left( t \right)} \right) < V\left( {x\left( {t_0 } \right)} \right) + \int\limits_{t_0 }^t {x*\left( \tau \right)Rx\left( \tau \right)d\tau ,} $$ where V(x) = x*Hx, is satisfied. This problem is solved for the cases where matrix A(·) has p sign-definite entries on the upper part of some subdiagonal or on the lower part of some superdiagonal. It is assumed also that all entries located to the left (or to the right) of the sign-definite entries are equal to zero.     

Non-isotropic hausdorff capacity of exceptional sets of invariant potentials

In the paper we investigate tangential boundary limits of invariant Green potentials on the unit ballB in ? ⁿ ,n≥1. LetG(z, w) denote the Green function for the Laplace-Beltrami operator onB, and let λ denote the invariant measure onB. If μ is a non-negative measure, orf is a non-negative measurable function onB,G _μ andG _f denote the Green potential of μ andf respectively. For ξ∈S=δB, τ≥1, andc>0, let $$\mathcal{T}_{\tau ,c} (\zeta ) = \{ z \in B:\left| {1 - \left\langle {z,\xi } \right\rangle } \right|^\tau< c(1 - \left| z \right|^2 )\} $$ . The main result of the paper is as follows: Letf be a non-negative measurable function onB satisfying $$\int_B {(1 - \left| w \right|^2 )^\beta f^p (w)d\lambda (w)< \infty } $$ for some β, 0<β<n, and somep>n. Then for each τ, 1≤τ<n/β, there exists a setE _t ?S withH _βτ (E _t )=0, such that $$\mathop {\lim }\limits_{\mathop {z \to \zeta }\limits_{z \in \mathcal{T}_{\tau ,c} (\zeta )} } G_f (z) = 0,forall\zeta \in S \sim E_\tau $$ In the above, for 0<α≤n,H _α denotes the non-isotropic α-dimensional Hausdorff capacity onS. We also prove that if {a _k } is a sequence inB satisfying Σ(1?|a _k |²) ^β <∞ for some β, 0 <β<n, and μ=Σδ _ak , where δ _a denotes point mass measure ata, then the same conclusion holds for the potentialG _μ .     

Commutators of the fractional maximal function on variable exponent Lebesgue spaces

Let \({M_\beta }\) be the fractional maximal function. The commutator generated by \({M_\beta }\) and a suitable function b is defined by \([{M_\beta },b]f = {M_\beta }(bf) - b{M_\beta }(f)\) . Denote by P(? ⁿ ) the set of all measurable functions p(·): ? ⁿ → [1,∞) such that $1 < p_ - : = \mathop {es\sin fp(x)}\limits_{x \in \mathbb{R}^n } andp_ + : = \mathop {es\operatorname{s} \sup p(x) < \infty }\limits_{x \in \mathbb{R}^n } ,$ and by B(? ⁿ ) the set of all p(·) ∈ P(? ⁿ ) such that the Hardy-Littlewood maximal function M is bounded on L ^p(·)(? ⁿ ). In this paper, the authors give some characterizations of b for which \([{M_\beta },b]\) is bounded from L ^p(·)(? ⁿ ) into L ^q(·)(? ⁿ ), when p(·) ∈ P(? ⁿ ), 0 < β < n/p ₊ and 1/q(·) = 1/p(·) ? β/n with q(·)(n ? β)/n ∈ B(? ⁿ ).     

On conditionally δ-convex functions

Let X be a real vector space, V a subset of X and δ ≧ 0 a given number. We say that f: V → ? is a conditionally δ-convex function if for each convex combination t ₁ υ ₁ + … + t _n υ _n of elements of V such that t ₁ υ ₁ + … + t _n υ _n ∈ V the following inequality holds true: $$ f(t_1 v_1 + \cdots + t_n v_n ) \leqq t_1 f(v_1 ) + \cdots + t_n f(v_n ) + \delta . $$ We prove that f: V → ? is conditionally δ-convex if and only if there exists a convex function $ \tilde f $ : conv V → [?∞, ∞) such that $$ \tilde f(v) \leqq f(v) \leqq \tilde f(v) + \delta for v \in V. $$ In case X = ? ⁿ some conditions equivalent to conditional δ-convexity are also presented.     

Limits of translates of divergent geodesics and integral points on one-sheeted hyperboloids

For any non-uniform lattice Γ in SL₂(?), we describe the limit distribution of orthogonal translates of a divergent geodesic in Γ\SL₂(?). As an application, for a quadratic form Q of signature (2, 1), a lattice Γ in its isometry group, and v ₀ ∈ ?³ with Q(v ₀) > 0, we compute the asymptotic (with a logarithmic error term) of the number of points in a discrete orbit v ₀Γ of norm at most T, when the stabilizer of v ₀ in Γ is finite. Our result in particular implies that for any non-zero integer d, the smoothed count for the number of integral binary quadratic forms with discriminant d ² and with coefficients bounded by T is asymptotic to c · T log T + O(T).