arXiv subjects
Publications and source records attributed to Ge Xiong.
For $n\geq2$ and $p\in(n-2,n)$, we prove that closed balls maximize the ratio of Riesz $p$-capacity to Newtonian capacity among all compact subsets of $\mathbb{R}^n$. This resolves the classical Pólya--Szegő capacity ratio conjecture and its higher-dimensional extensions. For $n\geq4$, a sharp inequality for the ratio of Newtonian capacity to logarithmic capacity is established. All equality cases are characterized.
The logarithmic Brunn--Minkowski and logarithmic Minkowski inequalities are proved for convex bodies whose centroids are at the origin, with equality precisely for independent positive dilations of common direct summands. This characterizes when centered convex bodies have the same cone-volume measure. The \(L_p\) Brunn--Minkowski and Minkowski inequalities are also obtained for every \(p>0\), with equality only for positive dilates, and the corresponding uniqueness of centered \(L_p\) surface area measures is established.
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 $m\geq 4$.
Let $μ_p$ 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 $α_p(n)$ be the largest number such that the Brunn-Minkowski type inequality $$μ_p(λK+(1-λ) L)^{α_p(n)} \geq λμ_p(K)^{α_p(n)}+(1-λ) μ_p(L)^{α_p(n)}$$ holds for all convex bodies $K,L$ in $\mathbb{R}^n$ containing the origin and $λ\in[0,1]$. In this paper, the new lower and upper bounds for $α_p(n)$ are found, and their asymptotically optimality as $n\to +\infty$ is proved.
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 $ρ\in L^{\infty}(\mathcal{S}^{n-1})$ satisfies $\mathcal{R}(ρ^i) = cρ$ for some $c>0$ iff $ρ$ is constant. Also, the sharp Busemann intersection type inequalities are established.
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.
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.
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.
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.
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.