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

• Record Nr.: UNINA9910966324603321
• Autore: Kolesnikov Alexander V
• Pubbl/distr/stampa: Providence : , : American Mathematical Society, , 2022; ©2022
• ISBN: 9781470470920; 1470470926
• Edizione: [1st ed.]
• Descrizione fisica: 1 online resource (90 pages)
• Collana: Memoirs of the American Mathematical Society ; ; v.277
• Classificazione: 52A4052A2335P1558J50
• Altri autori (Persone): MilmanEmanuel
• Disciplina: 516/.08; 516.08
• Soggetti: Convex domains; Lp spaces; Minkowski geometry; Inequalities (Mathematics); Convex and discrete geometry -- General convexity -- Inequalities and extremum problems; Convex and discrete geometry -- General convexity -- Asymptotic theory of convex bodies; Partial differential equations -- Spectral theory and eigenvalue problems -- Estimation of eigenvalues, upper and lower bounds; Global analysis, analysis on manifolds -- Partial differential equations on manifolds; differential operators -- Spectral problems; spectral geometry; scattering theory
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: 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.
• Sommario/riassunto: "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"--