Ισοπεριμετρικές ανισότητες για το μέτρο του Gauss

Παρουσιάζουμε γεωμετρικές ανισότητες για το μέτρο του Gauss στον n-διάστατο Ευκλείδειο χώρο. Η ισοπεριμετρική ανισότητα στον χώρο του Gauss ισχυρίζεται ότι ανάμεσα σε όλα τα Borel υποσύνολα του Rn που έχουν δεδομένο μέτρο α, οι ημίχωροι μέτρου α έχουν την ελάχιστη επιφάνεια. Παρουσιάζουμε τρεις αποδείξεις. Η πρώτη βασίζεται στην παρατήρηση του Poincare και ουσιαστικά ανάγει το πρόβλημα στο ισοπεριμετρικό πρόβλημα για την σφαίρα. Η δεύτερη βασίζεται στη μέθοδο της Gaussian συμμετρικοποίησης. Η τρίτη οφείλεται στον Bobkov και χρησιμοποιεί μια συναρτησιακή ανισότητα, η απόδειξη της οποίας, με τη σειρά της, βασίζεται σε μια ανισότητα δύο σημείων και στο κεντρικό οριακό θεώρημα, στο πνεύμα της αρχικής απόδειξης της λογαριθμικής ανισότητας Sobolev από τον Gross. Στη συνέχεια παρουσιάζουμε την απόδειξη της ανισότητας Ehrhard-Borell, η οποία είναι ισχυρότερη από την ισοπεριμετρική ανισότητα. Ο Ehrhard έδωσε μια απόδειξή της χρησιμοποιώντας τη μέθοδο της Gaussian συμμετρικοποίησης, όμως το επιχείρημά του περιοριζόταν στα κυρτά σύνολα. Τελικά, ο Borell αφαίρεσε την υπόθεση της κυρτότητας και απέδειξε την ανισότητα σε πλήρη γενικότητα. Περιγράφουμε το επιχείρημα του Ehrhard και το επιχείρημα του Borell το οποίο οδηγεί σε μια γενικότερη συναρτησιακή ανισότητα. Στο επόμενο κεφάλαιο μελετάμε το ρυθμό μεταβολής του μέτρου Gauss συμμετρικών κυρτών σωμάτων ως προς διαστολές. Το κεντρικό αποτέλεσμα οφείλεται στους Latala και Oleszkiewicz, οι οποίοι απέδειξαν μια εικασία του Shepp. Για την απόδειξη χρησιμοποιούμε αρχικά την ανισότητα του Ehrhard για να αναγάγουμε το πρόβλημα σε ένα τεχνικό αλλά διδιάστατο πρόβλημα. Στο επόμενο κεφάλαιο παρουσιάζουμε την πρόσφατη απόδειξη του Royen για την εικασία της θετικής συνδιακύμανσης για το μέτρο του Gauss: το μέτρο της τομής δύο συμμετρικών κυρτών σωμάτων είναι μεγαλύτερο ή ίσο από το γινόμενο των μέτρων τους. Στη συνέχεια παρουσιάζουμε το B-θεώρημα των Cordero-Erausquin, Fradelizi και Maurey, το οποίο απαντά θετικά σε μια εικασία του Banaszczyk. Η ανάλυση που γίνεται ανάγει το πρόβλημα σε μια ανισότητα τύπου Poincare για τον περιορισμό του μέτρου του Gauss σε ένα συμμετρικό κυρτό σώμα. Τέλος, παρουσιάζουμε εφαρμογές των παραπάνω γεωμετρικών ανισοτήτων σε γνωστά συνδυαστικά προβλήματα εξισορρόπησης διανυσμάτων.We present geometric inequalities for the Gaussian measure in n-dimensional Euclidean space. The isoperimetric inequality in Gauss space asserts that among all Borel subsets of Rn that have a given measure α, half-spaces of measure α have minimal Gaussian surface area. We present three proofs. The first one is based on an observation of Poincare and essentially reduces the problem to the isoperimetric problem for the sphere. The second one is based on the method of Gaussian symmetrization. The third one is due to Bobkov and employs a functional inequality, whose proof is in turn based on a two-point inequality and the central limit theorem, in the spirit of the original proof of the logarithmic Sobolev inequality by Gross. Then, we present the proof of the Ehrhard-Borell inequality, which is stronger than the isoperimetric inequality. Ehrhard gave a proof using Gaussian symmetrization, but his argument was restricted to the class of convex sets. Eventually, Borell removed the convexity assumption and proved the inequality in full generality. We describe the arguments of Ehrhard and Borell. The latter leads to a more general functional inequality. In the next chapter we study the behavior of the Gaussian measure of dilates of symmetric convex bodies. The main result is due to Latala and Oleszkiewicz, who confirmed a conjecture of Shepp. For the proof we use Ehrhard’s inequality to reduce the problem to a technical but two-dimensional problem. In the next chapter we present Royen’s recent proof of the Gaussian correlation conjecture: the measure of the intersection of two symmetric convex bodies is greater than or equal to the product of their measures. Next, we present the B-theorem of Cordero-Erausquin, Fradelizi and Maurey, which provides a positive answer to a conjecture of Banaszczyk. The analysis of the authors reduces the problem to a sharp inequality of We present geometric inequalities for the Gaussian measure in n-dimensional Euclidean space. The isoperimetric inequality in Gauss space asserts that among all Borel subsets of Rn that have a given measure α, half-spaces of measure α have minimal Gaussian surface area. We present three proofs. The first one is based on an observation of Poincare and essentially reduces the problem to the isoperimetric problem for the sphere. The second one is based on the method of Gaussian symmetrization. The third one is due to Bobkov and employs a functional inequality, whose proof is in turn based on a two-point inequality and the central limit theorem, in the spirit of the original proof of the logarithmic Sobolev inequality by Gross. Then, we present the proof of the Ehrhard-Borell inequality, which is stronger than the isoperimetric inequality. Ehrhard gave a proof using Gaussian symmetrization, but his argument was restricted to the class of convex sets. Eventually, Borell removed the convexity assumption and proved the inequality in full generality. We describe the arguments of Ehrhard and Borell. The latter leads to a more general functional inequality. In the next chapter we study the behavior of the Gaussian measure of dilates of symmetric convex bodies. The main result is due to Latala and Oleszkiewicz, who confirmed a conjecture of Shepp. For the proof we use Ehrhard’s inequality to reduce the problem to a technical but two-dimensional problem. In the next chapter we present Royen’s recent proof of the Gaussian correlation conjecture: the measure of the intersection of two symmetric convex bodies is greater than or equal to the product of their measures. Next, we present the B-theorem of Cordero-Erausquin, Fradelizi and Maurey, which provides a positive answer to a conjecture of Banaszczyk. The analysis of the authors reduces the problem to a sharp inequality of We present geometric inequalities for the Gaussian measure in n-dimensional Euclidean space. The isoperimetric inequality in Gauss space asserts that among all Borel subsets of Rn that have a given measure α, half-spaces of measure α have minimal Gaussian surface area. We present three proofs. The first one is based on an observation of Poincare and essentially reduces the problem to the isoperimetric problem for the sphere. The second one is based on the method of Gaussian symmetrization. The third one is due to Bobkov and employs a functional inequality, whose proof is in turn based on a two-point inequality and the central limit theorem, in the spirit of the original proof of the logarithmic Sobolev inequality by Gross. Then, we present the proof of the Ehrhard-Borell inequality, which is stronger than the isoperimetric inequality. Ehrhard gave a proof using Gaussian symmetrization, but his argument was restricted to the class of convex sets. Eventually, Borell removed the convexity assumption and proved the inequality in full generality. We describe the arguments of Ehrhard and Borell. The latter leads to a more general functional inequality. In the next chapter we study the behavior of the Gaussian measure of dilates of symmetric convex bodies. The main result is due to Latala and Oleszkiewicz, who confirmed a conjecture of Shepp. For the proof we use Ehrhard’s inequality to reduce the problem to a technical but two-dimensional problem. In the next chapter we present Royen’s recent proof of the Gaussian correlation conjecture: the measure of the intersection of two symmetric convex bodies is greater than or equal to the product of their measures. Next, we present the B-theorem of Cordero-Erausquin, Fradelizi and Maurey, which provides a positive answer to a conjecture of Banaszczyk. The analysis of the authors reduces the problem to a sharp inequality of Poincare-type for the restriction of the Gaussian measure onto a symmetric convex body. Finally, we present applications of these geometric inequalities to well-known combinatorial problems from discrepancy theory

A doubly exponential upper bound on noisy EPR states for binary games

This paper initiates the study of a class of entangled games, mono-state games, denoted by $(G,\psi)$, where $G$ is a two-player one-round game and $\psi$ is a bipartite state independent of the game $G$. In the mono-state game $(G,\psi)$, the players are only allowed to share arbitrary copies of $\psi$. This paper provides a doubly exponential upper bound on the copies of $\psi$ for the players to approximate the value of the game to an arbitrarily small constant precision for any mono-state binary game $(G,\psi)$, if $\psi$ is a noisy EPR state, which is a two-qubit state with completely mixed states as marginals and maximal correlation less than $1$. In particular, it includes $(1-\epsilon)|\Psi\rangle\langle\Psi|+\epsilon\frac{I_2}{2}\otimes\frac{I_2}{2}$, an EPR state with an arbitrary depolarizing noise $\epsilon>0$.The structure of the proofs is built the recent framework about the decidability of the non-interactive simulation of joint distributions, which is completely different from all previous optimization-based approaches or "Tsirelson's problem"-based approaches. This paper develops a series of new techniques about the Fourier analysis on matrix spaces and proves a quantum invariance principle and a hypercontractive inequality of random operators. This novel approach provides a new angle to study the decidability of the complexity class MIP$^*$, a longstanding open problem in quantum complexity theory.Comment: The proof of Lemma C.9 is corrected. The presentation is improved. Some typos are correcte

On the Relationship between Convex Bodies Related to Correlation Experiments with Dichotomic Observables

In this paper we explore further the connections between convex bodies related to quantum correlation experiments with dichotomic variables and related bodies studied in combinatorial optimization, especially cut polyhedra. Such a relationship was established in Avis, Imai, Ito and Sasaki (2005 J. Phys. A: Math. Gen. 38 10971-87) with respect to Bell inequalities. We show that several well known bodies related to cut polyhedra are equivalent to bodies such as those defined by Tsirelson (1993 Hadronic J. S. 8 329-45) to represent hidden deterministic behaviors, quantum behaviors, and no-signalling behaviors. Among other things, our results allow a unique representation of these bodies, give a necessary condition for vertices of the no-signalling polytope, and give a method for bounding the quantum violation of Bell inequalities by means of a body that contains the set of quantum behaviors. Optimization over this latter body may be performed efficiently by semidefinite programming. In the second part of the paper we apply these results to the study of classical correlation functions. We provide a complete list of tight inequalities for the two party case with (m,n) dichotomic observables when m=4,n=4 and when min{m,n}<=3, and give a new general family of correlation inequalities.Comment: 17 pages, 2 figure