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Gangsong Leng

Publications and source records attributed to Gangsong Leng.

15 recordsLinked to original sources
arXiv2026

A square-root complex inequality and its induced metric structure

Let $(\Omega,\mu)$ be a finite measure space with $M=\mu(\Omega)>0$. We investigate the integral form, stability, and metric geometry associated with a square-root complex. After proving the inequality and determining all equality cases, we analyze its phase stability near the intersection of the two branches of the equality set. In general phase directions, the quadratic term is precisely a Cauchy--Schwarz deficit; along the corresponding degenerate cone, the leading term is of fourth order and is strictly positive. A symmetric two-point example shows that the exponent four is unavoidable in any uniform distance-stability estimate. Finally, on the group of measurable circle-valued functions, we introduce the LY-metric \[ d_\mu(f,g)=\left|M-\int_\Omega f\overline g\,d\mu\right|^{1/2}. \] We prove that this metric is bi-invariant and complete, and that it induces the same topology as the $L^2$ metric. On finite-dimensional tori, we establish the optimality of the exponent $1/2$, derive explicit formulas for the intrinsic distance and geodesics, describe the anisotropic geometry and volume growth of small metric balls, and show that the Hausdorff dimension is $n+1$.

math.FA↗
arXiv2026

Distance Stability for the Integral Hardy and Carleman Inequalities

We study stability forms of two classical integral inequalities: Hardy's integral inequality and the integral Carleman inequality, also known as the P\'olya--Knopp inequality. The sharp constants in these two inequalities are $(p')^p$ and $e$, respectively, but neither inequality has a non-trivial extremizer in its natural function space. Their formal extremal functions are $c x^{-1/p}$ and $c/x$, respectively. We first derive a deficit identity for Hardy's integral inequality. In particular, when $p=2$, the deficit is exactly a squared norm. We then prove a distance stability estimate for the integral Carleman inequality, whose remainder term directly measures a local weighted Hellinger-type distance from the family $\{c/x:c\ge0\}$. Together these results illustrate the same stability phenomenon: although the classical integral inequalities have no genuine extremizers, their deficits still measure the deviation from the corresponding families of virtual extremals.

math.FA↗
arXiv2026

Two Integral Sliding-Window Inequalities for Maximal Convolutions

We prove two sliding-window inequalities for maximal convolutions. The first concerns the multiplicative maximal convolution. If $f$ and $g$ are nonnegative continuous functions on $[0,A]$ and $[0,B]$, respectively, define \[ h(x)=\max_{\substack{0\le u\le A\\0\le x-u\le B}} f(u)g(x-u),\qquad 0\le x\le A+B. \] Then there exists a window $[a,a+B]$ of length $B$ such that \[ \frac1B\int_a^{a+B}h(x)\,dx\ge \left(\frac1A\int_0^A f(x)\,dx\right) \left(\frac1B\int_0^B g(x)\,dx\right). \] The second concerns the additive maximal convolution. Let $f$ and $g$ be nonnegative continuous functions on $[0,C]$, and define \[ H(x)=\max_{\substack{0\le u\le C\\0\le x-u\le C}}\{f(u)+g(x-u)\},\qquad 0\le x\le 2C. \] Then, for every $p\ge1$, there exists a window $[a,a+C]$ of length $C$ such that \[ \left(\int_a^{a+C}H(x)^p\,dx\right)^{1/p} \ge \left(\int_0^C f(x)^p\,dx\right)^{1/p} + \left(\int_0^C g(x)^p\,dx\right)^{1/p}. \] We also record discrete analogues. The main point is that, in one-dimensional maximal-convolution settings, certain global Brunn--Minkowski or Pr\'ekopa--Leindler type phenomena admit natural sliding-window localizations.

math.FA↗
arXiv2026

A Density-Distance Version of the Carlen--Frank--Lieb Stability Theorem

Carlen, Frank and Lieb studied stability estimates for the lowest eigenvalue of a Schr\"odinger operator by decomposing the problem into a stability estimate for H\"older's inequality and a stability estimate for a Gagliardo--Nirenberg--Sobolev inequality. In this note we point out that, if the H\"older step is replaced by the optimal $L^1$-stability theorem of Leng and Lu in probabilistic form, then one obtains a density-distance version of the Carlen--Frank--Lieb stability theorem. The new formulation measures the $L^1$ distance between the normalized density $V_-^s/\int V_-^s$ induced by the negative part of the potential and the corresponding density induced by an optimal potential, where $s=\gamma+d/2$. As a geometric application of the same idea, we also derive a density-stability version of the $L_p$ mixed volume inequality. In the case where one of the two convex bodies is centrally symmetric and both bodies are trapped between two concentric Euclidean balls, this gives an averaged stability estimate for the non-evenness of the support function.

math.AP↗
arXiv2026

An Optimal Stability Theorem for H\"older's Inequality

We prove an optimal $L^1$ stability theorem for H\"older's inequality. Let $p>1$, $q>1$, and $1/p+1/q=1$. If $a_k,b_k\ge 0$ and \[ \sum_{k=1}^n a_k=\sum_{k=1}^n b_k=1, \] then \[ 1-\sum_{k=1}^n a_k^{1/p}b_k^{1/q} \ge \frac1{2pq}\left(\sum_{k=1}^n |a_k-b_k|\right)^2 . \] The constant $1/(2pq)$ is best possible. We also give the corresponding integral form.

math.FA↗
arXiv2026

A Median Version of Hardy's Inequality

Motivated by a discrete inequality problem proposed by Duanyang Zhang as Problem 6 of the 2022 Spring NSMO, we prove a median version of Hardy's inequality. For a nonnegative function $f\in L^p(0,\infty)$, $p>1$, let $A(t)$ be the average of $f$ over $(0,t)$, and let $M(t)$ be the lower median of $f$ over $(0,t)$. We show that \[ \int_0^\infty |M(t)-A(t)|^p\,dt \leq 2^{1-p}\left(\frac p{p-1}\right)^p \int_0^\infty f(t)^p\,dt, \] and that the constant is best possible. The proof is based on a pointwise rearrangement estimate coming from the half-measure property of the median, followed by the classical Hardy inequality. A discrete form and its sharpness are also included.

math.MG↗
arXiv2026

Inequalities on a Class of Function Sets

We prove a functional extension of an exponential inequality originally proposed by Bin Zhao and proved by Xiaosheng Mou. The main result asserts that if $\alpha_1\leq \cdots\leq \alpha_n$ and $\sum_{k=1}^n \alpha_k=0$, then \[ \sum_{k=1}^n \phi(k\alpha_k)\geq 0 \] for every odd function $\phi$ that is increasing and convex on $[0,\infty)$. The proof is based on a truncated-sum comparison and the stop-loss characterization of the increasing convex order. As consequences, we recover the original exponential inequality and obtain polynomial and integral variants.

math.FA↗
arXiv2024

$\operatorname{SL}(n)$ contravariant function-valued valuations on polytopes

We present a complete classification of $\operatorname{SL}(n)$ contravariant, $C(\mathbb{R}^n\setminus\{o\})$-valued valuations on polytopes, without any additional assumptions.It extends the previous results of the second author [Int. Math. Res. Not. 2020] which have a good connection with the $L_p$ and Orlicz Brunn-Minkowski theory. Additionally, our results deduce a complete classification of $\operatorname{SL}(n)$ contravariant symmetric-tensor-valued valuations on polytopes.

math.MG↗
arXiv2022

Dualities and endomorphisms of pseudo-cones

In this paper we study a class of convex sets which are called closed pseudo-cones and study a new duality of this class. It turns out that the duality characterizes closed pseudo-cones and is essentially the only possible abstract duality of them.The characterization of the duality is corresponding to the classification of endomorphisms closed pseudo-cones.

math.MG↗
arXiv2018

$L_p$ Minkowski Valuations on polytopes

For $1 \leq p < \infty$, Ludwig, Haberl and Parapatits classified $L_p$ Minkowski valuations intertwining the special linear group with additional conditions such as homogeneity and continuity. In this paper,a complete classification of $L_p$ Minkowski valuations intertwining the special linear group on polytopes without any additional conditions is established for $p \geq 1$ including $p = \infty$. For $n=3$ and $p=1$, there exist valuations not mentioned before.

math.MG↗
arXiv2018

$L_p$-Blaschke Valuations

In this article, a classification of continuous, linearly intertwining, symmetric $L_p$-Blaschke ($p>1$) valuations is established as an extension of Haberl's work on Blaschke valuations. More precisely, we show that for dimensions $n\geq 3$, the only continuous, linearly intertwining, normalized symmetric $L_p$-Blaschke valuation is the normalized $L_p$-curvature image operator, while for dimension $n = 2 $, a rotated normalized $L_p$-curvature image operator is an only additional one. One of the advantages of our approach is that we deal with normalized symmetric $L_p$-Blaschke valuations, which makes it possible to handle the case $p=n$. The cases where $p \not =n$ are also discussed by studying the relations between symmetric $L_p$-Blaschke valuations and normalized ones.

math.MG↗
arXiv2015

Orlicz valuations

In this paper, Orlicz valuations compatible with $SL(n)$ transforms are classified. Unlike their $L_p$ analogs, the identity operator and the reflection operator are the only $SL(n)$ compatible Orlicz valuations (up to dilations). It turns out that the Orlicz projection body operator, the Orlicz centroid body operator and the Orlicz difference body operator are not Orlicz valuations. The property that the Orlicz difference body operator is not an Orlicz valuation plays an important role in characterizing the identity operator and the reflection operator.

math.MG↗
arXiv2014

A new approach to Steiner symmetrization of coercive convex functions

In this paper, a new approach of defining Steiner symmetrization of coercive convex functions is proposed and some fundamental properties of the new Steiner symmetrization are proved. Further, using the new Steiner symmetrization, we give a different approach to prove a functional version of the Blaschke-Santalo inequality due to Ball.

math.DG↗
arXiv2014

Origin-Symmetric Bodies of Revolution with Minimal Mahler Volume in R^3-a new proof

Meyer and Reisner had proved the Mahler conjecture for rovelution bodies. In this paper, using a new method, we prove that among origin-symmetric bodies of revolution in R^3, cylinders have the minimal Mahler volume. Further, we prove that among parallel sections homothety bodies in R^3, 3-cubes have the minimal Mahler volume.

math.DG↗