arXiv subjects

Ge Xiong

Publications and source records attributed to Ge Xiong.

8 recordsLinked to original sources

arXiv2026

An affirmative solution to the generalized Busemann--Petty problem with subspace dimensions $2$ and $3$

The generalized Busemann--Petty problem asks whether origin-symmetric convex bodies in $\mathbb{R}^n$ having larger volume of all $m$-dimensional sections necessarily have larger volume. When $m\geq 4$, this is known to be false, but the cases $m=2, 3$ for $n\geq 5$ have remained open since the 1990s. In this paper, we resolve these cases. Together with the known results, the generalized Busemann--Petty problem is completely solved: the answer is affirmative for $m=1, 2, 3$, and negative for $m\geq 4$.

math.FA↗

arXiv2026

The Brunn-Minkowski inequality for the generalized Gaussian distribution

Let $\mu_p$ be the generalized Gaussian distribution on $\mathbb{R}^n$ with density $e^{-\frac{|x|^p}{p}}$ multiplied by a constant depending on $p\ge 1$ and $n$, and $\alpha_p(n)$ be the largest number such that the Brunn-Minkowski type inequality $$\mu_p(\lambda K+(1-\lambda) L)^{\alpha_p(n)} \geq \lambda \mu_p(K)^{\alpha_p(n)}+(1-\lambda) \mu_p(L)^{\alpha_p(n)}$$ holds for all convex bodies $K,L$ in $\mathbb{R}^n$ containing the origin and $\lambda\in[0,1]$. In this paper, the new lower and upper bounds for $\alpha_p(n)$ are found, and their asymptotically optimality as $n\to +\infty$ is proved.

math.MG↗

arXiv2025

Fixed and periodic points of the intersection body operators of lower orders

For the intersection body operator of lower order $I_iK$ of a star body $K$ in $\mathbb{R}^n$, $i\in\{1, 2,\ldots, n-2\}$, we prove that $I_i^2K = cK$ iff $K$ is an origin-symmetric ball, and hence $I_iK = cK$ iff $K$ is an origin-symmetric ball. Combining the recent breakthrough (case $i = n-1$) of Milman, Shabelman and Yehudayoff (Invent. Math., 241 (2025), 509-558), slight modifications of two long-standing questions 8.6 and 8.7 posed by R. Gardner (Page 302, Geometric Tomography, Cambridge University Press, 1995) are completely solved. As applications, we show that for the spherical Radon transform $\mathcal{R}$, a non-negative $\rho\in L^{\infty}(\mathcal{S}^{n-1})$ satisfies $\mathcal{R}(\rho^i) = c\rho$ for some $c>0$ iff $\rho$ is constant. Also, the sharp Busemann intersection type inequalities are established.

math.MG↗

arXiv2024

A matroid polytope approach to sharp affine isoperimetric inequalities for volume decomposition functionals

New sharp affine isoperimetric inequalities for volume decomposition functionals $X_{2}$ and $X_{3}$ in $\mathbb{R}^n$ are established. To fulfil this task, we prove the recursion formulas for volume decomposition functionals and find out the connection between the domains of these functionals and matroid polytopes. Applications of matroid theory to convex geometry are presented.

math.MG↗

arXiv2022

The logarithmic Minkowski inequality for cylinders

In this paper, we prove that if $K$ is an $o$-symmetric cylinder and $L$ is an $o$-symmetric convex body in $\mathbb R^3$, then the logarithmic Minkowski inequality \[\frac{1}{V(K)}\int_{\mathbb S^{2}}\log\frac{h_L}{h_K}\,dV_K\geq\frac{1}{3}\log\frac{V(L)}{V(K)}\] holds, with equality if and only if $K$ and $L$ are relative cylinders.

math.MG↗

arXiv2017

Lp Minkowski problem for electrostatic $\mathfrak{p}$-capacity

Existence and uniqueness of the solution to the discrete Lp Minkowski problem for $\mathfrak{p}$-capacity are proved when $p \geq 1$ and $1<\mathfrak{p}<n$. For general Lp Minkowski problem for $\mathfrak{p}$-capacity, existence and uniqueness of the solution are given when $p \geq 1$ and $1<\mathfrak{p}\le 2$. These results are non-linear extensions of the very recent solution to the Lp Minkowski problem for $\mathfrak{p}$-capacity when $p=1$ and $1<\mathfrak{p}\le n$ by CNSXYZ, and the classical soution to the Minkowski problem for electrostatic capacity when $p=1$ and $\mathfrak{p}=2$ by Jerison.

math.DG↗

arXiv2016

A unified treatment for Lp Brunn-Minkowski type inequalities

A unified approach used to generalize classical Brunn-Minkowski type inequalities to Lp Brunn-Minkowski type inequalities, called the Lp transference principle, is refined in this paper. As illustrations of the effectiveness and practicability of this method, several new Lp Brunn-Minkowski type inequalities concerning the mixed volume, moment of inertia, quermassintegral, projection body and capacity are established.

math.MG↗

arXiv2014

Orlicz-Legendre Ellipsoids

The Orlicz-Legendre ellipsoids, which are in the framework of emerging dual Orlicz Brunn-Minkowski theory, are introduced for the first time. They are in some sense dual to the recently found Orlicz-John ellipsoids, and have largely generalized the classical Legendre ellipsoid of inertia. Several new affine isoperimetric inequalities are established. The connection between the characterization of Orlicz-Legendre ellipsoids and isotropy of measures is demonstrated.

math.MG↗