Supervisors info:
Απόστολος Γιαννόπουλος, Καθηγητής του Τμήματος Μαθηματικών, ΕΚΠΑ (επιβλέπων)
Γατζαούρας Δημήτριος, Καθηγητής του Τμήματος Μαθηματικών, ΕΚΠΑ
Χελιώτης Δημήτριος, Αναπληρωτής Καθηγητής του Τμήματος Μαθηματικών, ΕΚΠΑ
Summary:
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.
