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Über dieses Buch

As in the previous Seminar Notes, the current volume reflects general trends in the study of Geometric Aspects of Functional Analysis, understood in a broad sense. A classical theme in the Local Theory of Banach Spaces which is well represented in this volume is the identification of lower-dimensional structures in high-dimensional objects. More recent applications of high-dimensionality are manifested by contributions in Random Matrix Theory, Concentration of Measure and Empirical Processes. Naturally, the Gaussian measure plays a central role in many of these topics, and is also studied in this volume; in particular, the recent breakthrough proof of the Gaussian Correlation Conjecture is revisited. The interplay of the theory with Harmonic and Spectral Analysis is also well apparent in several contributions. The classical relation to both the primal and dual Brunn-Minkowski theories is also well represented, and related algebraic structures pertaining to valuations and valent functions are discussed. All contributions are original research papers and were subject to the usual refereeing standards.

Inhaltsverzeichnis

Frontmatter

On Repeated Sequential Closures of Constructible Functions in Valuations

Abstract
The space of constructible functions form a dense subspace of the space of generalized valuations. In this note we prove a somewhat stronger property that the sequential closure, taken sufficiently many (in fact, infinitely many) times, of the former space is equal to the latter one. This stronger property is necessary for some applications in Alesker (Geom Funct Anal 20(5):1073–1143, 2010).
Semyon Alesker

Orbit Point of View on Some Results of Asymptotic Theory; Orbit Type and Cotype

Abstract
We develop an orbit point of view on the notations of type and cotype and extend Kwapien’s theorem to this setting. We show that such approach provides an exact equality in the latter theorem. In addition, we discuss several well known theorems and reformulate them using the orbit point of view.
Limor Ben-Efraim, Vitali Milman, Alexander Segal

Concentration Properties of Restricted Measures with Applications to Non-Lipschitz Functions

Abstract
We show that, for any metric probability space (M, d, μ) with a subgaussian constant σ 2(μ) and any Borel measurable set A ⊂ M, we have \(\sigma ^{2}(\mu _{A}) \leq c\log \left (e/\mu (A)\right )\sigma ^{2}(\mu )\), where μ A is a normalized restriction of μ to the set A and c is a universal constant. As a consequence, we deduce concentration inequalities for non-Lipschitz functions.
Sergey G. Bobkov, Piotr Nayar, Prasad Tetali

On Random Walks in Large Compact Lie Groups

Abstract
Let G be the group SO(d) or SU(d) with d large. How long does it take for a random walk on G to approximate uniform measure? It is shown that in certain natural examples an ɛ-approximation is achieved in time \(\left (d\log \frac{1} {\varepsilon } \right )^{C}\).
Jean Bourgain

On a Problem of Farrell and Vershynin in Random Matrix Theory

Abstract
We settle a question of Farrell and Vershynin on the inverse of the perturbation of a given arbitrary symmetric matrix by a GOE element.
Jean Bourgain

Valuations on the Space of Quasi-Concave Functions

Abstract
We characterize the valuations on the space of quasi-concave functions on \(\mathbb{R}^{N}\), that are rigid motion invariant and continuous with respect to a suitable topology. Among them we also provide a specific description of those which are additionally monotone.
Andrea Colesanti, Nico Lombardi

An Inequality for Moments of Log-Concave Functions on Gaussian Random Vectors

Abstract
We prove sharp moment inequalities for log-concave and log-convex functions, on Gaussian random vectors. As an application we take a reverse form of the classical logarithmic Sobolev inequality, in the case where the function is log-concave.
Nikos Dafnis, Grigoris Paouris

-Valent Functions

Abstract
We introduce the notion of \((\mathcal{F},p)\)-valent functions. We concentrate in our investigation on the case, where \(\mathcal{F}\) is the class of polynomials of degree at most s. These functions, which we call (s, p)-valent functions, provide a natural generalization of p-valent functions (see Hayman, Multivalent Functions, 2nd ed, Cambridge Tracts in Mathematics, vol 110, 1994). We provide a rather accurate characterizing of (s, p)-valent functions in terms of their Taylor coefficients, through “Taylor domination”, and through linear non-stationary recurrences with uniformly bounded coefficients. We prove a “distortion theorem” for such functions, comparing them with polynomials sharing their zeroes, and obtain an essentially sharp Remez-type inequality in the spirit of Yomdin (Isr J Math 186:45–60, 2011) for complex polynomials of one variable. Finally, based on these results, we present a Remez-type inequality for (s, p)-valent functions.
Omer Friedland, Yosef Yomdin

A Remark on Projections of the Rotated Cube to Complex Lines

Abstract
Motivated by relations with a symplectic invariant known as the “cylindrical symplectic capacity”, in this note we study the expectation of the area of a minimal projection to a complex line for a randomly rotated cube.
Efim D. Gluskin, Yaron Ostrover

On the Expectation of Operator Norms of Random Matrices

Abstract
We prove estimates for the expected value of operator norms of Gaussian random matrices with independent (but not necessarily identically distributed) and centered entries, acting as operators from \(\ell_{p^{{\ast}}}^{n}\) to q m , 1 ≤ p  ≤ 2 ≤ q < .
Olivier Guédon, Aicke Hinrichs, Alexander E. Litvak, Joscha Prochno

The Restricted Isometry Property of Subsampled Fourier Matrices

Abstract
A matrix \(A \in \mathbb{C}^{q\times N}\) satisfies the restricted isometry property of order k with constant ε if it preserves the 2 norm of all k-sparse vectors up to a factor of 1 ±ε. We prove that a matrix A obtained by randomly sampling q = O(k ⋅ log2 k ⋅ logN) rows from an N × N Fourier matrix satisfies the restricted isometry property of order k with a fixed ε with high probability. This improves on Rudelson and Vershynin (Comm Pure Appl Math, 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).
Ishay Haviv, Oded Regev

Upper Bound for the Dvoretzky Dimension in Milman-Schechtman Theorem

Abstract
For a symmetric convex body \(K \subset \mathbb{R}^{n}\), the Dvoretzky dimension k(K) is the largest dimension for which a random central section of K is almost spherical. A Dvoretzky-type theorem proved by V.D. Milman in 1971 provides a lower bound for k(K) in terms of the average M(K) and the maximum b(K) of the norm generated by K over the Euclidean unit sphere. Later, V.D. Milman and G. Schechtman obtained a matching upper bound for k(K) in the case when \(\frac{M(K)} {b(K)}> c(\frac{\log (n)} {n} )^{\frac{1} {2} }\). In this paper, we will give an elementary proof of the upper bound in Milman-Schechtman theorem which does not require any restriction on M(K) and b(K).
Han Huang, Feng Wei

Super-Gaussian Directions of Random Vectors

Abstract
We establish the following universality property in high dimensions: Let X be a random vector with density in \(\mathbb{R}^{n}\). The density function can be arbitrary. We show that there exists a fixed unit vector \(\theta \in \mathbb{R}^{n}\) such that the random variable \(Y =\langle X,\theta \rangle\) satisfies
$$\displaystyle{ \min \left \{\mathbb{P}(Y \geq tM), \mathbb{P}(Y \leq -tM)\right \} \geq ce^{-Ct^{2} }\qquad \qquad \text{for all}\ 0 \leq t \leq \tilde{ c}\sqrt{n}, }$$
where M > 0 is any median of | Y | , i.e., \(\min \{\mathbb{P}(\vert Y \vert \geq M), \mathbb{P}(\vert Y \vert \leq M)\} \geq 1/2\). Here, \(c,\tilde{c},C> 0\) are universal constants. The dependence on the dimension n is optimal, up to universal constants, improving upon our previous work.
Bo’az Klartag

A Remark on Measures of Sections of -balls

Abstract
We prove that there exists an absolute constant C so that
$$\displaystyle{ \mu (K)\ \leq \ C\sqrt{p}\max _{\xi \in S^{n-1}}\mu (K \cap \xi ^{\perp })\ \vert K\vert ^{1/n} }$$
for any p > 2, any \(n \in \mathbb{N},\) any convex body K that is the unit ball of an n-dimensional subspace of L p , and any measure μ with non-negative even continuous density in \(\mathbb{R}^{n}.\) Here ξ  ⊥  is the central hyperplane perpendicular to a unit vector ξ ∈ S n−1, and | K | stands for volume.
Alexander Koldobsky, Alain Pajor

Sharp Poincaré-Type Inequality for the Gaussian Measure on the Boundary of Convex Sets

Abstract
A sharp Poincaré-type inequality is derived for the restriction of the Gaussian measure on the boundary of a convex set. In particular, it implies a Gaussian mean-curvature inequality and a Gaussian iso-second-variation inequality. The new inequality is nothing but an infinitesimal equivalent form of Ehrhard’s inequality for the Gaussian measure. While Ehrhard’s inequality does not extend to general CD(1, ) measures, we formulate a sufficient condition for the validity of Ehrhard-type inequalities for general measures on \(\mathbb{R}^{n}\) via a certain property of an associated Neumann-to-Dirichlet operator.
Alexander V. Kolesnikov, Emanuel Milman

Rigidity of the Chain Rule and Nearly Submultiplicative Functions

Abstract
Assume that \(T: C^{1}(\mathbb{R}) \rightarrow C(\mathbb{R})\) nearly satisfies the chain rule in the sense that
$$\displaystyle{\vert T(\,f \circ g)(x) - (Tf)(g(x))(Tg)(x)\vert \leq S(x,(\,f \circ g)(x),g(x))}$$
holds for all \(f,g \in C^{1}(\mathbb{R})\) and \(x \in \mathbb{R}\), where \(S: \mathbb{R}^{3} \rightarrow \mathbb{R}\) is a suitable fixed function. We show under a weak non-degeneracy and a weak continuity assumption on T that S may be chosen to be 0, i.e. that T satisfies the chain rule operator equation, the solutions of which are explicitly known. We also determine the solutions of one-sided chain rule inequalities like
$$\displaystyle{T(\,f \circ g)(x) \leq (Tf)(g(x))(Tg)(x) + S(x,(\,f \circ g)(x),g(x))}$$
under a further localization assumption. To prove the above results, we investigate the solutions of nearly submultiplicative inequalities on \(\mathbb{R}\)
$$\displaystyle{\phi (\alpha \beta ) \leq \phi (\alpha )\phi (\beta ) + d}$$
and characterize the nearly multiplicative functions on \(\mathbb{R}\)
$$\displaystyle{\vert \phi (\alpha \beta ) -\phi (\alpha )\phi (\beta )\vert \leq d\ }$$
under weak restrictions on ϕ.
Hermann König, Vitali Milman

Royen’s Proof of the Gaussian Correlation Inequality

Abstract
We present in detail Thomas Royen’s proof of the Gaussian correlation inequality which states that μ(KL) ≥ μ(K)μ(L) for any centered Gaussian measure μ on \(\mathbb{R}^{d}\) and symmetric convex sets K, L in \(\mathbb{R}^{d}\).
Rafał Latała, Dariusz Matlak

A Simple Tool for Bounding the Deviation of Random Matrices on Geometric Sets

Abstract
Let A be an isotropic, sub-gaussian m × n matrix. We prove that the process \(Z_{x}\,:=\,\left \|Ax\right \|_{2} -\sqrt{m}\left \|x\right \|_{2}\) has sub-gaussian increments, that is, \(\|Z_{x} - Z_{y}\|_{\psi _{2}} \leq C\|x - y\|_{2}\) for any \(x,y \in \mathbb{R}^{n}\). Using this, we show that for any bounded set \(T \subseteq \mathbb{R}^{n}\), the deviation of ∥ Ax ∥ 2 around its mean is uniformly bounded by the Gaussian complexity of T. We also prove a local version of this theorem, which allows for unbounded sets. These theorems have various applications, some of which are reviewed in this paper. In particular, we give a new result regarding model selection in the constrained linear model.
Christopher Liaw, Abbas Mehrabian, Yaniv Plan, Roman Vershynin

On Multiplier Processes Under Weak Moment Assumptions

Abstract
We show that if \(V \subset \mathbb{R}^{n}\) satisfies a certain symmetry condition that is closely related to unconditionality, and if X is an isotropic random vector for which \(\|\big< X,t\big >\| _{L_{p}} \leq L\sqrt{p}\) for every t ∈ S n−1 and every \(1 \leq p\lesssim \log n\), then the suprema of the corresponding empirical and multiplier processes indexed by V behave as if X were L-subgaussian.
Shahar Mendelson

Characterizing the Radial Sum for Star Bodies

Abstract
In this paper we prove two theorems characterizing the radial sum of star bodies. By doing so we demonstrate an interesting phenomenon: essentially the same conditions, on two different spaces, can uniquely characterize very different operations. In our first theorem we characterize the radial sum by its induced homothety, and our list of assumptions is identical to the assumptions of the corresponding theorem which characterizes the Minkowski sum for convex bodies. In our second theorem give a different characterization from a short list of natural properties, without assuming the homothety has any specific form. For this theorem one has to add an assumption to the corresponding theorem for convex bodies, as we demonstrate by a simple example.
Vitali Milman, Liran Rotem

On Mimicking Rademacher Sums in Tail Spaces

Abstract
We establish upper and lower bounds for the L 1 distance from a Rademacher sum to the mth tail space on the discrete cube. The bounds are tight, up to the value of multiplicative constants.
Krzysztof Oleszkiewicz

Stability for Borell-Brascamp-Lieb Inequalities

Abstract
We study stability issues for the so-called Borell-Brascamp-Lieb inequalities, proving that when near equality is realized, the involved functions must be L 1-close to be p-concave and to coincide up to homotheties of their graphs.
Andrea Rossi, Paolo Salani

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