---
title: Zhang's Inequality in Geometry and Analysis
url: https://www.emergentmind.com/topics/zhang-s-inequality
type: topic
---

# Zhang's Inequality in Geometry and Analysis

Zhang's inequality is not a single universally fixed theorem. In the arXiv literature, the expression is used for several distinct statements associated with authors named Zhang: most prominently a reverse affine projection inequality for convex bodies, but also affine Sobolev inequalities, determinant refinements, matrix norm inequalities, operator inequalities, and a recent sharp median analogue of Hardy's inequality motivated by Duanyang Zhang’s 2022 Spring NSMO Problem 6. This suggests that the name is context-dependent, and that precise meaning is determined by the surrounding subject, notation, and extremal class [1810.07507], [2506.10473], [2007.13155], [2605.25366].

## 1. Classical affine-geometric meaning

In convex geometry, Zhang's inequality is a reverse affine projection inequality. For a convex body \(K\subseteq \mathbb{R}^n\), it states that among all convex bodies, the simplex minimizes the affine invariant quantity
\[
(\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),
\]
where \(\Pi^*K\) is the polar projection body. One normalization used in the literature is
\[
\frac{2n}{\binom{2n}{n}n^n}\le (\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),
\]
with equality attained for simplices [2509.14986].

The same inequality is also presented as the left-hand side of a two-sided affine isoperimetric inequality. In the notation of one cited paper,
\[
\frac{1}{n^n}\binom{2n}{n} \le |K|^{\,n-1}\,|\Pi^\circ K| \le \left(\frac{\kappa_n}{\kappa_{n-1}}\right)^n,
\]
where the left-hand side is called Zhang’s projection inequality and the right-hand side is Petty’s projection inequality; equality on the Zhang side holds if and only if \(K\) is an \(n\)-simplex, while equality on the Petty side holds if and only if \(K\) is an ellipsoid [2304.07859].

A central structural point is that the inequality is affine invariant and simplex-extremized. The cited literature also uses both \(\Pi^*K\) and \(\Pi^\circ K\) for the polar projection body, reflecting notational differences rather than a change in substance [1810.07507].

## 2. Functional, higher-order, and discrete extensions

A functional extension replaces convex bodies by integrable log-concave functions. For \(f\in\mathcal{F}(\mathbb{R}^n)\), the functional polar projection body \(\Pi^*(f)\) is defined from the level sets
\[
K_t(f)=\{x\in\mathbb{R}^n:f(x)\ge e^{-t}\|f\|_\infty\},
\]
and the main inequality takes the form
\[
\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\min\{f(x),f(y)\}\,dx\,dy \le 2n\,n!\,\|f\|_\infty\,|\Pi^*(f)|.
\]
If \(\|f\|_\infty=f(0)\), equality holds if and only if
\[
f(x)=e^{-\|x\|_A}
\]
for some \(n\)-dimensional simplex \(A\) containing the origin [1810.07507].

A second functional route uses a Berwald-type monotonicity principle on epigraphs of log-concave functions. For an integrable log-concave \(f:\mathbb{R}^n\to[0,\infty)\) and a concave function \(h\) on
\[
L=\{(x,t)\in\mathbb{R}^{n+1}: f(x)\ge e^{-t}\|f\|_\infty\},
\]
the map
\[
p \mapsto \left(\frac{1}{\Gamma(1+p)\int_L e^{-t}\,dt\,dx}\int_L h^p(x,t)e^{-t}\,dt\,dx\right)^{1/p}
\]
is decreasing for \(p\in(-1,\infty)\). This extension is used to give a new proof of a functional form of Zhang’s reverse Petty projection inequality, with equality characterized by simplex-type extremizers \(f(x)=e^{-\|x\|_{A_n}}\) [1908.01154].

Higher-order extensions replace the usual covariogram and projection body by \(m\)-fold analogues. The resulting higher-order Zhang inequality is
\[
|K|^{nm-m}\,|\mathbb{PP}K| \ge \frac{1}{n^{nm}\binom{nm+n}{n}},
\]
with equality if and only if \(K\) is an \(n\)-dimensional simplex. The case \(m=1\) recovers the classical Zhang inequality [2304.07859].

The same geometric theme also has a limiting-convolution-body formulation. In that setting, the case \(k=n-1\) is described as the exact analogue of Zhang’s inequality, and the sharp equality condition is
\[
K=-L \quad\text{and this common body is a simplex}
\]
for the relevant volume inequality involving \(C_{n-1}(K,L)\) [1312.6005].

A discrete approach replaces Lebesgue measure by lattice-point counting. With the lattice point enumerator
\[
G_k(A)=|A\cap \mathbb{Z}^k|
\]
and a mixed measure \(d\mu=dG_{n-1}\otimes dm_1\), discrete analogues of the covariogram inequality and the associated projection-body inclusion are proved; by scaling \(K\mapsto \lambda K\) and sending \(\lambda\to\infty\), the continuous inequality, and therefore Zhang’s inequality, is recovered [2509.14986].

## 3. Affine Sobolev forms and Zhang's energy

A different major usage is Zhang’s affine Sobolev inequality. In one formulation, for an integrable \(p\)-concave function \(f\),
\[
|\pp f| \leq |\pp f^*|,
\]
where \(f^*\) is the symmetric decreasing rearrangement and \(\pp f\) denotes the polar projection body of the function. The same paper proves a stochastic version in expectation for random models built from \(p\)-concave functions and then derives the deterministic generalization
\[
\nu(\pp f) \leq \nu(\pp f^*)
\]
for any rotationally invariant convex measure \(\nu\) on \(\mathbb{R}^n\) [2509.04108].

In homogeneous Sobolev spaces, Zhang’s refinement replaces the classical gradient norm by a smaller affine-invariant quantity. The higher-order generalization defines, for \(s>0\) and \(1\le p<\infty\) with \(sp<N\), an affine functional \(\mathscr{E}_{s,p}(f)\) based on directional higher differences or higher directional derivatives, satisfying
\[
\mathscr{E}_{s,p}(f\circ T)=\mathscr{E}_{s,p}(f), \qquad T\in \mathrm{SL}_N,
\]
and yields the affine embedding
\[
\|f\|_{L^{Np/(N-sp)}(\mathbb{R}^N)} \leq K_{s,p,N}\,\mathscr{E}_{s,p}(f).
\]
For \(s=1\), this recovers Zhang’s original affine-Sobolev structure [2506.10473].

On \(BV(\Omega)\), the relevant object is Zhang’s affine \(L^1\) energy. For \(u\in BV(\mathbb{R}^n)\),
\[
E_{\mathbb{R}^n}(u) = a_n \left(\int_{S^{n-1}} \left(\int_{\mathbb{R}^n} | \sigma_u(x)\cdot \xi |\, d|Du|(x)\right)^{-n} d\xi\right)^{-1/n},
\]
with \(Du=\sigma_u\,d|Du|\). This functional is affine invariant,
\[
E_{\mathbb{R}^n}(u\circ T)=E_{\mathbb{R}^n}(u)\qquad \forall\,T\in SL(n),
\]
and satisfies
\[
E_{\mathbb{R}^n}(u)\le |Du|(\mathbb{R}^n).
\]
It underlies sharp affine Poincaré, Poincaré-Wirtinger, and Poincaré-Sobolev inequalities on \(BV(\Omega)\), and the cited paper proves existence of extremals in the subcritical range [2111.12347].

## 4. Linear-algebraic and matrix-analytic meanings

In matrix theory, Zhang’s inequality may denote the Zhang–Yang sharpening of Hadamard’s determinant inequality. For an \(n\times n\) Hermitian positive semidefinite matrix \(A=(a_{ij})\) and a nontrivial permutation \(\sigma\),
\[
\det(A)+\prod_{i=1}^n |a_{i,\sigma(i)}| \le \prod_{i=1}^n a_{ii}.
\]
The refinement is proved in the cited work by a majorization argument, and for positive definite matrices the equality conditions are completely characterized [2007.13155].

Another usage concerns sectorial matrices. If \(A\) is partitioned as
\[
A=\begin{bmatrix}A_{11}&A_{21}\\ A_{12}&A_{22}\end{bmatrix}
\]
and \(W(A)\subseteq S_\alpha\), then P. Zhang’s Rotfel’d-type inequality compares \(\|f(|A|)\|\) with block-diagonal terms for concave \(f:[0,\infty)\to[0,\infty)\). A 2024 refinement introduces a free parameter \(s>0\) and strengthens Zhang’s theorem by means of the block positivity criterion
\[
\begin{bmatrix}
\sec\alpha\,R_A & A^*\\
A & \sec\alpha\,R_A
\end{bmatrix}\ge 0
\]
for sectorial matrices [2404.12849].

The name also appears in generalized matrix-function inequalities. For positive semidefinite matrices \(A,B,C\), Zhang et al. proved
\[
d_\chi(A+B+C)+d_\chi(A)+d_\chi(B)+d_\chi(C)
\ge
d_\chi(A+B)+d_\chi(A+C)+d_\chi(B+C),
\]
and a later paper extends this to positive semidefinite block matrices through partial generalized matrix functions \(d_{\chi1}\) and \(d_{\chi2}\) [2003.05230].

A further matrix-analytic meaning is Teng Zhang’s hybrid triangle inequality. For \(X_k\in M_d\),
\[
\left|\sum_{k=1}^m X_k\right| \prec_w \sqrt{2}\sum_{k=1}^m |X_k|_{qsym},
\]
where
\[
|Z|_{qsym}:=\sqrt{\frac{|Z|^2+|Z^*|^2}{2}}.
\]
The constant \(\sqrt2\) is stated to be optimal in every dimension \(d\ge 2\). The 2026 refinement derives operator and eigenvalue versions from a new polar decomposition for the quadratic symmetric modulus [2606.29188].

## 5. Number-theoretic and arithmetic usages

In analytic number theory, the phrase appears in work surrounding bounded gaps between primes. In that setting, the relevant estimate is the deduction
\[
\liminf_{n\to\infty}(p_{n+1}-p_n)\le \operatorname{diam}(\mathcal H)
\]
from Zhang’s theorem on admissible \(k_0\)-tuples. Trudgian used this step with an admissible block of consecutive primes to improve the explicit prime-gap bound from \(70\) million to \(59{,}874{,}594\) [1305.6369].

In arithmetic dynamics, the relevant named object is Zhang’s fundamental inequality. The cited 2025 paper describes the classical form as comparing the canonical height of a subvariety to the essential minimum of points on that subvariety, and proves a quantitative dynamical refinement: sufficiently small points are contained in a proper exceptional subvariety whose degree admits an explicit bound. The result is then applied to Bogomolov-type gap principles for Néron–Tate heights and regular polynomial endomorphisms of \(\mathbb{P}^2\) [2512.07655].

These usages are mathematically unrelated to the affine-geometric and matrix-theoretic forms. Their coexistence under the same label reinforces the context-sensitive character of the term.

## 6. Median Hardy inequality motivated by Duanyang Zhang

A recent and sharply formulated use of Zhang’s name arises from a discrete inequality problem proposed by Duanyang Zhang as Problem 6 of the 2022 Spring NSMO. For a nonnegative function \(f\in L^p(0,\infty)\), \(p>1\), let
\[
A(t)=\frac1t\int_0^t f(s)\,ds
\]
be the average over \((0,t)\), and let \(M(t)\) be the lower median
\[
M(t)=\inf\left\{a\in\mathbb R: \operatorname{meas}\{s\in(0,t):f(s)\leq a\}\geq \frac t2\right\}.
\]
The main theorem is
\[
\int_0^\infty |M(t)-A(t)|^p\,dt \leq 2^{1-p}\left(\frac p{p-1}\right)^p \int_0^\infty f(t)^p\,dt,
\]
and the constant is best possible [2605.25366].

The proof proceeds from the pointwise rearrangement estimate
\[
|M(t)-A(t)|\leq \frac1t\int_0^{t/2} f^*(s)\,ds,
\]
where \(f^*\) is the decreasing rearrangement, followed by the classical Hardy inequality. The factor \(2^{1-p}\) comes from the half-measure property of the median, and the factor \(\left(\frac p{p-1}\right)^p\) is the classical best constant in Hardy’s inequality [2605.25366].

The paper also proves the discrete analogue: for \(x_1,\dots,x_n\ge 0\), with
\[
a_i=\frac{x_1+\cdots+x_i}{i}
\]
and \(m_i\) the lower median of \(x_1,\dots,x_i\),
\[
\sum_{i=1}^n |m_i-a_i|^p \leq 2^{1-p}\left(\frac p{p-1}\right)^p\sum_{i=1}^n x_i^p.
\]
For \(p=2\), the constant becomes \(2\), matching Zhang’s original discrete problem. This places the Olympiad-style question inside a broader \(L^p\) Hardy framework and gives a sharp continuous-discrete correspondence [2605.25366].

Source: https://www.emergentmind.com/topics/zhang-s-inequality