arXiv subjects

Trí Minh Lê

Publications and source records attributed to Trí Minh Lê.

8 recordsLinked to original sources

arXiv2026

Sharp regularity results for eikonal equations in Banach spaces

Viscosity solutions of eikonal equations need not enjoy regularity beyond Lipschitz continuity. In contrast, differentiable solutions exhibit stronger structural properties. In the Euclidean setting, building on observations of Caffarelli and Crandall (Comm. Partial Differential Equations $\textbf{35}$ (2010), 391--414), one finds that differentiable solutions of the eikonal equation $\| Df \| \equiv 1 \,\, \text{in } \mathcal{U} \subset \mathbb{R}^d$ are of class $C^{1, 1}_{\mathrm{loc}}(\mathcal U)$. Moreover, if $\mathcal{U} = \mathbb{R}^d$, then $f$ must be affine. This manuscript revisits these phenomena in the setting of (finite-- and infinite--dimensional) Banach spaces. Our results show how the geometry of the underlying Banach space determines the regularity and affineness of Gateaux--differentiable solutions of eikonal equations. As a geometric application in finite dimensions, we characterize differentiable functions whose gradient ranges are constrained to boundaries of convex bodies.

math.AP↗

arXiv2026

A convexity criterion via the De Giorgi slope

Let $X$ be a Banach space and $f\in\mathcal{C}^1(X)$ be bounded from below. We show that if for some $m\geq 1$, the function $x\mapsto \|\nabla f(x)\|^m$ is convex, then $f$ is convex. We also establish a more general version of this result: if $f$ is continuous and bounded from below, then it is convex, provided $x\mapsto s_f(x)^m$ is convex for some $m\geq 1$, where $s_f$ denotes the (De Giorgi) metric slope of $f$.

math.FA↗

arXiv2026

Impossibility of a nontrivial Brunn--Minkowski inequality for higher Dirichlet eigenvalue

Let $λ_j(K)$ be the $j$th Dirichlet eigenvalue of a convex body $K$. It is well known that $λ_1$ satisfies a Brunn--Minkowski inequality: $K \mapsto λ_1(K)^{-1/2}$ is concave on the family of convex bodies. We show that no analogous statement holds for higher eigenvalues. More precisely, for any $j \geq 2$ and $N \geq 2$, if $K \mapsto (f \circ λ_j)(K)$ is concave on the family of convex bodies in $\mathbb{R}^N$ for some function $f: (0, \infty) \to \mathbb{R}$, then $f$ must be constant.

math.MG↗

arXiv2026

On the growth of nonconvex functionals at strict local minimizers

We give new characterizations of growth conditions at strict local minimizers. The main characterizations are a variant of the so-called tilt stability property and an analog of the classical Polyak--Łojasiewicz condition, where the gradient is replaced by linear perturbations.

math.OC↗

arXiv2026

A note on absolutely minimal extensions in finite metric spaces

Absolutely minimal Lipschitz extensions (AMLEs) are known to exist in many infinite metric settings, but the finite case is less settled. In metric spaces with at most four points, every function on a nonempty subset admits an AMLE in the sense that the Lipschitz constant cannot be further reduced on sets that are disjoint from the prescribed domain. We show that in five-point spaces such extensions may fail to exist.

math.MG↗

arXiv2026

Power-law asymptotics of fractional $L^p$ polar projection bodies

The notion of $s$--fractional $L^p$ polar projection bodies, recently introduced by Haddad and Ludwig (Math.\ Ann.\ \textbf{388}:1091--1115, 2024), provides a bridge between fractional Sobolev theory and convex geometry. In this manuscript, we study the limit of their Minkowski gauges under two natural asymptotic regimes: \\ \hspace*{3em} (a) first sending $p \to \infty$ and then $s \to 1^-$; \\ \hspace*{3em} (b) first sending $s \to 1^-$ and then $p \to \infty$. \\ Our main result shows that these two limiting processes commute. As a consequence, we derive precise asymptotic behavior for the associated volumes and dual mixed volumes, thereby linking the (fractional) $L^p$ polar projection bodies to the newly introduced (fractional) $L^\infty$ polar projection bodies. These results further yield new geometric inequalities, including endpoint Lipschitz/Hölder isoperimetric--type and variants of Pólya--Szegő inequalities in the $L^\infty$ setting.

math.MG↗

arXiv2025

Computing intrinsic volumes of sublevel sets and applications

Intrinsic volumes are fundamental geometric invariants generalizing volume, surface area, and mean width for convex bodies. We establish a unified Laplace-Grassmannian representation for intrinsic and dual volumes of convex polynomial sublevel sets. More precisely, let $f$ be a convex $d$-homogeneous polynomial of even degree $d \ge 2$ which is positive except at the origin. We show that the intrinsic and dual volumes of the sublevel set $[f \le 1]$ admit Laplace-type integral formulas obtained by averaging the infimal projection and restriction of $f$ over the Grassmannian. This explicit representation yields three main consequences: (1) Löwner--John-type existence and uniqueness results extending beyond the classical volume case; (2) a block decomposition principle describing factorization of intrinsic volumes under direct-sum splitting; (3) a coordinate-free formulation of Lipschitz-type lattice discrepancy bounds. These formulas enable analytic treatment of a broad class of geometric quantities, providing direct access to variational and arithmetic applications as well as new structural insights.

math.MG↗

arXiv2024

On (discounted) global Eikonal equations in metric spaces

Eikonal equations in metric spaces have strong connections with the local slope operator (or the De Giorgi slope). In this manuscript, we explore and delve into an analogous model based on the global slope operator, expressed as $λu + G[u] = \ell$, where $λ\geq 0$. In strong contrast with the classical theory, the global slope operator relies neither on the local properties of the functions nor on the structure of the space, and therefore new insights are developed in order to analyze the above equation. Under mild assumptions on the metric space $X$ and the given data $\ell$, we primarily discuss: $(a)$ the existence and uniqueness of (pointwise) solutions; $(b)$ a viscosity perspective and the employment of Perron's method to consider the maximal solution; $(c)$ stability of the maximal solution with respect to both, the data $\ell$ and the discount factor $λ$. Our techniques provide a method to approximate solutions of Eikonal equations in metric spaces and a new integration formula based on the global slope of the given function.

math.AP↗