The logarithmic Minkowski conjecture and the L<sub>p</sub>-Minkowski problem

Böröczky, Károly J.

doi:10.1515/9783110775389-003

Harmonic Analysis and Convexity 2023

DOI: 10.1515/9783110775389-003

|View full text |Cite

The logarithmic Minkowski conjecture and the L_p-Minkowski problem

Károly J. Böröczky¹

Abstract: The current state of art concerning the L p -Minkowski problem as a Monge-Ampére equation on the sphere and Lutwak's Logarithmic Minkowski conjecture about the uniqueness of even solution in the p = 0 case are surveyed and connections to many related problems are discussed.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2023

2024

Publication Types

Select...

Article4

Relationship

Self Cite0

Independent4

Authors

Journals

Cited by 4 publications

References 200 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Variational Worn Stones

Crasta,

Fragalà

2024

Arch Rational Mech Anal

View full text Add to dashboard Cite

We introduce an evolution model à la Firey for a convex stone which tumbles on a beach and undertakes an erosion process depending on some variational energy, such as torsional rigidity, a principal Dirichlet Laplacian eigenvalue, or Newtonian capacity. Relying on the assumption of the existence of a solution to the corresponding parabolic flow, we prove that the stone tends to become asymptotically spherical. Indeed, we identify an ultimate shape of these flows with a smooth convex body whose ground state satisfies an additional boundary condition, and we prove symmetry results for the corresponding overdetermined elliptic problems. Moreover, we extend the analysis to arbitrary convex bodies: we introduce new notions of cone variational measures and we prove that, if such a measure is absolutely continuous with constant density, the underlying body is a ball.

show abstract

Variational Worn Stones

Crasta,

Fragalà

2024

Arch Rational Mech Anal

View full text Add to dashboard Cite

show abstract

Uniqueness when the $$L_p$$ curvature is close to be a constant for $$p\in [0,1)$$

Böröczky,

Saroglou

2024

Calc. Var.

View full text Add to dashboard Cite

For fixed positive integer n, $$p\in [0,1)$$ p ∈ [ 0 , 1 ) , $$a\in (0,1)$$ a ∈ ( 0 , 1 ) , we prove that if a function $$g:{\mathbb {S}}^{n-1}\rightarrow {\mathbb {R}}$$ g : S n - 1 → R is sufficiently close to 1, in the $$C^a$$ C a sense, then there exists a unique convex body K whose $$L_p$$ L p curvature function equals g. This was previously established for $$n=3$$ n = 3 , $$p=0$$ p = 0 by Chen et al. (Adv Math 411(A):108782, 2022) and in the symmetric case by Chen et al. (Adv Math 368:107166, 2020). Related, we show that if $$p=0$$ p = 0 and $$n=4$$ n = 4 or $$n\le 3$$ n ≤ 3 and $$p\in [0,1)$$ p ∈ [ 0 , 1 ) , and the $$L_p$$ L p curvature function g of a (sufficiently regular, containing the origin) convex body K satisfies $$\lambda ^{-1}\le g\le \lambda $$ λ - 1 ≤ g ≤ λ , for some $$\lambda >1$$ λ > 1 , then $$\max _{x\in {\mathbb {S}}^{n-1}}h_K(x)\le C(p,\lambda )$$ max x ∈ S n - 1 h K ( x ) ≤ C ( p , λ ) , for some constant $$C(p,\lambda )>0$$ C ( p , λ ) > 0 that depends only on p and $$\lambda $$ λ . This also extends a result from Chen et al. [10]. Along the way, we obtain a result, that might be of independent interest, concerning the question of when the support of the $$L_p$$ L p surface area measure is lower dimensional. Finally, we establish a strong non-uniqueness result for the $$L_p$$ L p -Minkowksi problem, for $$-n<p<0$$ - n < p < 0 .

show abstract

On subspace concentration for dual curvature measures

Eller

Henk

2023

Advances in Applied Mathematics

View full text Add to dashboard Cite

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

The logarithmic Minkowski conjecture and the L_p-Minkowski problem

Abstract: The current state of art concerning the L p -Minkowski problem as a Monge-Ampére equation on the sphere and Lutwak's Logarithmic Minkowski conjecture about the uniqueness of even solution in the p = 0 case are surveyed and connections to many related problems are discussed.

Cited by 4 publications

References 200 publications

Variational Worn Stones

Variational Worn Stones

Uniqueness when the $$L_p$$ curvature is close to be a constant for $$p\in [0,1)$$

On subspace concentration for dual curvature measures

Contact Info

Product

Resources

About

The logarithmic Minkowski conjecture and the Lp-Minkowski problem

Abstract: The current state of art concerning the L p -Minkowski problem as a Monge-Ampére equation on the sphere and Lutwak's Logarithmic Minkowski conjecture about the uniqueness of even solution in the p = 0 case are surveyed and connections to many related problems are discussed.

Cited by 4 publications

References 200 publications

Variational Worn Stones

Variational Worn Stones

Uniqueness when the $$L_p$$ curvature is close to be a constant for $$p\in [0,1)$$

On subspace concentration for dual curvature measures

Contact Info

Product

Resources

About

The logarithmic Minkowski conjecture and the L_p-Minkowski problem