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Mohammad N. Ivaki

Publications and source records attributed to Mohammad N. Ivaki.

At least 19 recordsLinked to original sources
arXiv2026

Centro-affine spectral geometry of polytopes

In this paper, we develop a polyhedral version of Milman's centro-affine spectral framework for full-dimensional origin-symmetric polytopes. A weighted graph Laplacian on the facets plays the role of the smooth centro-affine Laplacian. For $n\geq 3$, we prove that its first nonconstant even eigenvalue satisfies \[ λ_{1,e}(P)\geq n-1+\frac{1}{n-1} \] whenever $P$ is a local minimizer of $λ_{1,e}$ among origin-symmetric polytopes with the same facet normals, but not necessarily the same normal fan.

math.MG↗
arXiv2026

Weighted centro-affine Poincaré inequalities

We obtain weighted centro-affine Bochner formulas on spherical caps associated with smooth strictly convex hypersurfaces. As a consequence, we prove weighted Poincaré inequalities on caps and on intersections of caps for a class of weights depending on the position vector $X$ of the hypersurface. In the unconditional case, we obtain a centro-affine Poincaré inequality with weight $|X|^2$, which is used to prove a Brunn--Minkowski inequality for the $(n+2)$-nd dual quermassintegral. We also establish an $L_0$-Brunn--Minkowski inequality for the $q$-th dual quermassintegral for $q\in(0,n)$, with equality only for dilates, and an $L_p$-Brunn--Minkowski inequality for $q=n+α$ whenever \[ 0<α\le \frac{2p(1-p)}{2-p}, \] which in particular covers the range $q\in(n,n+6-4\sqrt{2}]$ for suitable $p\in(0,1)$. These Brunn--Minkowski inequalities imply weighted centro-affine Poincaré inequalities and uniqueness results for the $L_{p,q}$-Minkowski problem in the unconditional class. Our main contribution is the introduction of a flat logarithmic centro-affine geometry on the positive orthant $(0,\infty)^n$, adapted to the multiplicative structure of the $L_0$-sum. In this geometry, a Bochner formula yields a sharp Poincaré inequality, as well as a new proof of the centro-affine Poincaré inequality with constant $n$ due to Kolesnikov--Milman, for unconditional bodies and unconditional functions.

math.AP↗
arXiv2026

On the conjectured capillary Blaschke-Santaló inequality

We prove that the conjectured capillary Blaschke--Santaló inequality holds for any unconditional, strictly convex capillary hypersurface when $θ\in \left(0, \tfracπ{2}\right)$. Moreover, for $θ\in \left(\tfracπ{2}, π\right)$, we show that the capillary volume product has no finite upper bound.

math.DG↗
arXiv2026

Capillary $L_p$-curvature problem

We prove a gradient estimate for a class of capillary curvature equations in the half-space. As an application, we prove the existence of an even, smooth, strictly convex solution to the even capillary $L_p$-curvature problem for all $1<p<k+1$ and all contact angles $θ\in(0,π/2)$.

math.AP↗
arXiv2025

Capillary $L_p$-Christoffel-Minkowski problem

We solve the capillary $L_p$-Christoffel--Minkowski problem in the half-space for $1<p<k+1$ in the class of even hypersurfaces. A crucial ingredient is a non-collapsing estimate that yields lower bounds for both the height and the capillary support function. Our result extends the capillary Christoffel--Minkowski existence result of \cite{HIS25}.

math.AP↗
arXiv2025

Capillary $L_p$ Minkowski Flows

We study the long-time existence and asymptotic behavior of a class of anisotropic capillary Gauss curvature flows. As an application, we provide a flow approach to the existence of smooth solutions to the capillary even $L_p$ Minkowski problem in the Euclidean half-space for all $p \in (-n-1, \infty)$ and capillary $L_p$ Minkowski problem for $p > n+1$.

math.AP↗
arXiv2025

Capillary curvature images

In this paper, we solve the even capillary $L_p$-Minkowski problem for the range $-n < p < 1$ and $θ\in (0,\fracπ{2})$. Our approach is based on an iterative scheme that builds on the solution to the capillary Minkowski problem (i.e., the case $p = 1$) and leverages the monotonicity of a class of functionals under a family of capillary curvature image operators. These operators are constructed so that their fixed points, whenever they exist, correspond precisely to solutions of the capillary $L_p$-Minkowski problem.

math.DG↗
arXiv2025

Capillary Christoffel-Minkowski problem

The result of Guan and Ma (Invent. Math. 151 (2003)) states that if $ϕ^{-1/k} : \mathbb{S}^n \to (0,\infty)$ is spherically convex, then $ϕ$ arises as the $σ_k$ curvature (the $k$-th elementary symmetric function of the principal radii of curvature) of a strictly convex hypersurface. In this paper, we establish an analogous result in the capillary setting in the half-space for $θ\in(0,π/2)$: if $ϕ^{-1/k} : \mathcal{C}_θ \to (0,\infty)$ is a capillary function and spherically convex, then $ϕ$ is the $σ_k$ curvature of a strictly convex capillary hypersurface.

math.DG↗
arXiv2024

New quermassintegral and Poincaré type inequalities for non-convex domains

In the first part of this paper, we study the following non-homogeneous, locally constrained inverse curvature flow in Euclidean space $\mathbb{R}^{n+1}$, \begin{align*} \dot{x}=\left(\frac{1}{\frac{E_k(\hatκ)}{E_{k-1}(\hatκ)}-α}-\langle x,ν\rangle\right)ν, \quad k=2,3,\ldots,n-1. \end{align*} Assuming that the initial hypersurface $\mathcal{M}_0 \subset \mathbb{R}^{n+1}$ is star-shaped and its shifted principal curvatures $\hatκ=κ+α(1,\ldots,1)$ lie in the convex set \begin{align*} Γ_{α,k}:=Γ_{k-1}\cap \{λ\in \mathbb{R}^n:\, E_k(λ)-αE_{k-1}(λ)>0\}, \end{align*} we show that the flow admits a smooth solution that exists for all positive times, and it converges smoothly to a round sphere. As a corollary, we obtain a new set of Alexandrov-Fenchel-type inequalities for non-convex domains. In the second part, we derive a Poincaré type inequality for $k$-convex hypersurfaces which complements a more general version of the well-known Heintze-Karcher inequality.

math.DG↗
arXiv2024

Prescribed $L_p$ curvature problem

In this paper, we establish the existence of smooth, origin-symmetric, strictly convex solutions to the prescribed even $L_p$ curvature problem.

math.AP↗
arXiv2023

$L^p$-Minkowski Problem under Curvature Pinching

Let $K$ be a smooth, origin-symmetric, strictly convex body in $\mathbb{R}^n$. If for some $\ell\in GL(n,\mathbb{R})$, the anisotropic Riemannian metric $\frac{1}{2}D^2 \Vert\cdot\Vert_{\ell K}^2$, encapsulating the curvature of $\ell K$, is comparable to the standard Euclidean metric of $\mathbb{R}^{n}$ up-to a factor of $γ> 1$, we show that $K$ satisfies the even $L^p$-Minkowski inequality and uniqueness in the even $L^p$-Minkowski problem for all $p \geq p_γ:= 1 - \frac{n+1}γ$. This result is sharp as $γ\searrow 1$ (characterizing centered ellipsoids in the limit) and improves upon the classical Minkowski inequality for all $γ< \infty$. In particular, whenever $γ\leq n+1$, the even log-Minkowski inequality and uniqueness in the even log-Minkowski problem hold.

math.DG↗
arXiv2023

Uniqueness of solutions to a class of non-homogeneous curvature problems

We show that the only even, smooth, convex solutions to a class of isotropic mixed Christoffel-Minkowski type problems are origin-centred spheres, which, in particular, answers a question of Firey 74 in the even isotropic case about kinematic measures. Employing the Heintze-Karcher inequality, we prove that the only smooth, strictly convex solutions to a large class of Minkowski type problems are origin-centred spheres. Immediate corollaries are the uniqueness of solutions to the isotropic Orlicz-Minkowski problem and the isotropic $L_p$-Gaussian-Minkowski problem when $p\geq 1$.

math.DG↗
arXiv2023

Uniqueness of solutions to a class of isotropic curvature problems

Employing a local version of the Brunn-Minkowski inequality, we give a new and simple proof of a result due to Andrews, Choi and Daskalopoulos that the origin-centred balls are the only closed, self-similar solutions of the Gauss curvature flow. Extensions to various non-linearities are obtained, assuming the centroid of the enclosed convex body is at the origin. By applying our method to the Alexandrov-Fenchel inequality, we also show that origin-centred balls are the only solutions to a large class of even Christoffel-Minkowski type problems.

math.DG↗
arXiv2022

On the stability of the $L_p$-curvature

It is known that the $L_p$-curvature of a smooth, strictly convex body in $\mathbb{R}^{n}$ is constant only for origin-centred balls when $1\neq p>-n$, and only for balls when $p=1$. If $p=-n$, then the $L_{-n}$-curvature is constant only for origin-symmetric ellipsoids. We prove `local' and `global' stability versions of these results. For $p\geq 1$, we prove a global stability result: if the $L_p$-curvature is almost a constant, then the volume symmetric difference of $\tilde{K}$ and a translate of the unit ball $B$ is almost zero. Here $\tilde{K}$ is the dilation of $K$ with the same volume as the unit ball. For $0\leq p<1$, we prove a similar result in the class of origin-symmetric bodies in the $L^2$-distance. In addition, for $-n 2$ in the Banach-Mazur distance.

math.MG↗
arXiv2020

Constant rank theorems for curvature problems via a viscosity approach

An important set of theorems in geometric analysis consists of constant rank theorems for a wide variety of curvature problems. In this paper, for geometric curvature problems in compact and non-compact settings, we provide new proofs which are both elementary and short. Moreover, we employ our method to obtain constant rank theorems for homogeneous and non-homogeneous curvature equations in new geometric settings. One of the essential ingredients for our method is a generalization of a differential inequality in a viscosity sense satisfied by the smallest eigenvalue of a linear map (Brendle-Choi-Daskalopoulos, Acta Math. 219(2017): 1-16) to the one for the subtrace. The viscosity approach provides a concise way to work around the well known technical hurdle that eigenvalues are only Lipschitz in general. This paves the way for a simple induction argument.

math.AP↗