arXiv subjects

Zhe-Feng Xu

Publications and source records attributed to Zhe-Feng Xu.

8 recordsLinked to original sources

arXiv2026

Optimal Transport and the ABP Method in Higher Codimension

We prove an inverse optimal-transport principle for the ABP contact set of animmersed submanifold. A fixed function on the submanifold determines, for every probability density on a bounded convex target, a source measure and an optimal plan. For a uniform ellipsoidal target, disintegration over the submanifold gives conditional measures whose fiberwise $L^\infty$-norms satisfy a sharp weighted average estimate with constant $ω_n/ω_{n+m}$ in codimensions $1\leq m\leq4$. This yields the sharp ellipsoidal Michael--Simon--Sobolev inequality in the same range, including its equality cases and symmetrization consequences. To the best of our knowledge, even in the Euclidean case, the sharp constants in codimensions three and four were previously unknown.

math.DG↗

arXiv2026

Ideal points, directed completion and the case of the maximally extended Schwarzschild spacetime

We study the directed completion of Lorentzian pre-length spaces, and we show that, under some natural assumptions which cover the case of smooth globally hyperbolic spacetimes, it coincides with the future causal completion of Geroch--Kronheimer--Penrose. Moreover, we provide applications of our findings by characterizing the directed completion of the Kruskal--Szekeres spacetime in terms of its radial null geodesics.

math.DG↗

arXiv2026

The many-body Blaschke-Santaló type inequality via optimal transport

Let $K_1,\ldots,K_k\subset\mathbb R^n$ be origin-symmetric measurable sets of finite volume such that \[ \sum_{1\le i<j\le k}\langle x_i,x_j\rangle\le \binom{k}{2}, \qquad \forall\,x_i\in K_i, x_j\in K_j. \] We prove the sharp many-body Blaschke--Santaló type inequality \[ \prod_{i=1}^k |K_i|\le |B^n|^k \] proposed by Kalantzopoulos and Saroglou, and characterize all equality cases. The proof combines multi-marginal optimal transport with a pseudo-Euclidean volume estimate. Using the geometric--functional equivalence of Kalantzopoulos and Saroglou, we also establish the functional version inequality proposed by Kolesnikov and Werner.

math.AP↗

arXiv2026

Total curvature and length estimates for timelike curves in Lorentzian length spaces

We introduce and study a synthetic notion of timelike total curvature for curves in Lorentzian length spaces with upper curvature bounds. In particular, we prove that our notion agrees with its smooth counterpart, and we show that timelike curves of finite total curvature are rectifiable. As the main application, we provide a sharp lower bound for the length of timelike curves solely in terms of the time separation between their endpoints and their total curvature.

math.DG↗

arXiv2026

The Mahler Conjecture in Three Dimensions

The Mahler conjecture dates back to 1938. This paper solves the conjecture for general convex bodies in three dimensions by developing a method called the shadow flow. The equality case is characterized as well. This method is also applied to give a new proof of the three-dimensional symmetric case, which was first proved by Iriyeh--Shibata.

math.MG↗

arXiv2026

A new characterization of Sobolev spaces on Lipschitz differentiability spaces

Numerous characterizations of Sobolev norms via the asymptotic behavior of non-local functionals have been established over the past decades; however, their validity beyond the PI framework remains poorly understood. We establish such a characterization on Lipschitz differentiability spaces without assuming either the doubling condition or a Poincaré inequality, by proving sharp two-sided Brezis--Van Schaftingen--Yung type asymptotic formulas. We also construct sharp counterexamples revealing the necessity of our assumptions, and provide several examples which are of independent interest.

math.FA↗

arXiv2026

$L^2$-Caffarelli--Kohn--Nirenberg inequalities on metric measure spaces

Motivated by the sharp constants in the $L^2$-Caffarelli--Kohn--Nirenberg (or $L^2$-CKN for short) inequalities on Euclidean spaces, we study, in a unified framework, a sequence of $L^2$-CKN inequalities on metric measure spaces. On a general metric measure space, this sequence implies a reverse volume comparison of Günther type. Moreover, on a subclass of spaces admitting the measure contraction property, we show that this sequence of $L^2$-CKN inequalities are valid if and only if the spaces are volume cones. We also provide a stability result for inequalities of this type on volume cones.

math.FA↗

arXiv2026

Lorentz meets Ptolemy

We consider a Lorentzian analogue of the Ptolemy inequality and we prove that in the setting of globally hyperbolic spacetimes it is equivalent to a global timelike sectional curvature bound from above by zero. We investigate the link between the Ptolemy inequality and the hyperbolic inversion and establish some applications and rigidity properties.

math.DG↗