Local Lp-Brunn-Minkowski inequalities for p < 1 / Alexander V. Kolesnikov, Emanuel Milman.

Format:

Book

Author/Creator:

Kolesnikov, Alexander V.

Contributor:

Milman, Emanuel.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.277

Language:

English

Subjects (All):

Convex domains.

Lp spaces.

Minkowski geometry.

Inequalities (Mathematics).

Physical Description:

1 online resource (90 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2022.

Summary:

"The Lp-Brunn-Minkowski theory for p<1, proposed by Firey and developed by Lutwak in the 90's, replaces the Minkowski addition of convex sets by its Lp counterpart, in which the support functions are added in Lp-norm. Recently, Boroczky, Lutwak, Yang and Zhang have proposed to extend this theory further to encompass the range. In particular, they conjectured an Lp-Brunn-Minkowski inequality for origin-symmetric convex bodies in that range, which constitutes a strengthening of the classical Brunn-Minkowski inequality. Our main result confirms this conjecture locally for all (smooth) origin-symmetric convex bodies in Rn and. In addition, we confirm the local log-Brunn-Minkowski conjecture (the case ) for small-enough C2-perturbations of the unit-ball of for q 2, when the dimension n is sufficiently large, as well as for the cube, which we show is the conjectural extremal case. For unit-balls of with q, we confirm an analogous result for , a universal constant. It turns out that the local version of these conjectures is equivalent to a minimization problem for a spectral-gap parameter associated with a certain differential operator, introduced by Hilbert (under different normalization) in his proof of the Brunn- Minkowski inequality. As applications, we obtain local uniqueness results in the even Lp-Minkowski problem, as well as improved stability estimates in the Brunn- Minkowski and anisotropic isoperimetric inequalities"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. Previously Known Partial Results

1.2. Main Results

1.3. Spectral Interpretation via the Hilbert-Brunn-Minkowski operator

1.4. Method of Proof

1.5. Applications

Chapter 2. Notation

Chapter 3. Global vs. Local Formulations of the ^{ }-Brunn-Minkowski Conjecture

3.1. Standard Equivalent Global Formulations

3.2. Global vs. Local ^{ }-Brunn-Minkowski

Chapter 4. Local ^{ }-Brunn-Minkowski Conjecture -Infinitesimal Formulation

4.1. Mixed Surface Area and Volume of ² functions

4.2. Properties of Mixed Surface Area and Volume

4.3. Second ^{ }-Minkowski Inequality

4.4. Comparison with classical =1 case

4.5. Infinitesimal Formulation On ⁿ⁻¹

4.6. Infinitesimal Formulation On ∂

Chapter 5. Relation to Hilbert-Brunn-Minkowski Operator and Linear Equivariance

5.1. Hilbert-Brunn-Minkowski operator

5.2. Linear equivariance of the Hilbert-Brunn-Minkowski operator

5.3. Spectral Minimization Problem and Potential Extremizers

Chapter 6. Obtaining Estimates via the Reilly Formula

6.1. A sufficient condition for confirming the local -BM inequality

6.2. General Estimate on \D( )

6.3. Examples

Chapter 7. The second Steklov operator and \B( ₂ⁿ)

7.1. Second Steklov operator

7.2. Computing \B( ₂ⁿ)

Chapter 8. Unconditional Convex Bodies and the Cube

8.1. Unconditional Convex Bodies

8.2. The Cube

Chapter 9. Local log-Brunn-Minkowski via the Reilly Formula

9.1. Sufficient condition for verifying local log-Brunn-Minkowski

9.2. An alternative derivation via estimating \B( )

Chapter 10. Continuity of \B, \BNH, \D with application to _{ }ⁿ

10.1. Continuity of \B, \BNH, \D in -topology

10.2. The Cube

10.3. Unit-balls of ℓ_{ }ⁿ

Chapter 11. Local Uniqueness for Even ^{ }-Minkowski Problem.

Chapter 12. Stability Estimates for Brunn-Minkowski and Isoperimetric Inequalities

12.1. New stability estimates for origin-symmetric convex bodies with respect to variance

12.2. Improved stability estimates for all convex bodies with respect to asymmetry

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

9781470470920

1470470926

OCLC:

1343250800