On $L_p$ Brunn-Minkowski type inequalities for a general class of functionals
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909525233106944 |
|---|---|
| author | Malagón, Lidia Gordo Nicolás, Jesús Yepes |
| author_facet | Malagón, Lidia Gordo Nicolás, Jesús Yepes |
| contents | In this work, the $L_p$ version (for $p> 1$) of the dimensional Brunn-Minkowski inequality for the standard Gaussian measure $γ_n(\cdot)$ on $\mathbb{R}^n$ is shown. More precisely, we prove that for any $0$-symmetric convex sets with nonempty interior, any $p>1$, and every $λ\in (0,1)$, \[ γ_n\bigl((1-λ)\cdot K+_p λ\cdot L\bigr)^{p/n} \geqslant (1-λ) γ_n(K)^{p/n} + λγ_n(L)^{p/n}, \] with equality, for some $λ\in (0,1)$ and $p>1$, if and only if $K=L$. This result, recently established without the equality conditions by Hosle, Kolesnikov and Livshyts, by using a different and functional approach, turns out to be the $L_p$ extension of a celebrated result for the Minkowski sum (that is, for $p=1$) by Eskenazis and Moschidis (2021) on a problem by Gardner and Zvavitch (2010).
Moreover, an $L_p$ Brunn-Minkowski type inequality is obtained for the classical Wills functional $\mathcal{W}(\cdot)$ of convex bodies.
These results are derived as a consequence of a more general approach, which provides us with other remarkable examples of functionals satisfying $L_p$ Brunn-Minkowski type inequalities, such as different absolutely continuous measures with radially decreasing densities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00153 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $L_p$ Brunn-Minkowski type inequalities for a general class of functionals Malagón, Lidia Gordo Nicolás, Jesús Yepes Metric Geometry Functional Analysis Primary 52A40, 28A25, Secondary 52A20, 52A39, 26B15 In this work, the $L_p$ version (for $p> 1$) of the dimensional Brunn-Minkowski inequality for the standard Gaussian measure $γ_n(\cdot)$ on $\mathbb{R}^n$ is shown. More precisely, we prove that for any $0$-symmetric convex sets with nonempty interior, any $p>1$, and every $λ\in (0,1)$, \[ γ_n\bigl((1-λ)\cdot K+_p λ\cdot L\bigr)^{p/n} \geqslant (1-λ) γ_n(K)^{p/n} + λγ_n(L)^{p/n}, \] with equality, for some $λ\in (0,1)$ and $p>1$, if and only if $K=L$. This result, recently established without the equality conditions by Hosle, Kolesnikov and Livshyts, by using a different and functional approach, turns out to be the $L_p$ extension of a celebrated result for the Minkowski sum (that is, for $p=1$) by Eskenazis and Moschidis (2021) on a problem by Gardner and Zvavitch (2010). Moreover, an $L_p$ Brunn-Minkowski type inequality is obtained for the classical Wills functional $\mathcal{W}(\cdot)$ of convex bodies. These results are derived as a consequence of a more general approach, which provides us with other remarkable examples of functionals satisfying $L_p$ Brunn-Minkowski type inequalities, such as different absolutely continuous measures with radially decreasing densities. |
| title | On $L_p$ Brunn-Minkowski type inequalities for a general class of functionals |
| topic | Metric Geometry Functional Analysis Primary 52A40, 28A25, Secondary 52A20, 52A39, 26B15 |
| url | https://arxiv.org/abs/2503.00153 |