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Table of 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( ₂ ₂₀₇ )
7.1. Second Steklov operator
7.2. Computing \B( ₂ ₂₀₇ )
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.
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( ₂ ₂₀₇ )
7.1. Second Steklov operator
7.2. Computing \B( ₂ ₂₀₇ )
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.