Zhang's Inequality in Geometry and Analysis

• Zhang's inequality is a context-dependent concept that, in convex geometry, establishes a reverse affine projection inequality with simplex extremizers.
• Extensions include functional forms for log-concave functions, higher-order and discrete analogues, and matrix-analytic refinements of classical inequalities.
• Recent applications cover affine Sobolev inequalities, sharpening Hadamard’s determinant bounds, and a median Hardy inequality with optimal constants.

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 (Alonso-Gutiérrez et al., 2018, Bullion-Gauthier, 12 Jun 2025, Lin et al., 2020, Leng, 25 May 2026).

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 (Alonso-Gutiérrez et al., 18 Sep 2025).

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 (Haddad et al., 2023).

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 (Alonso-Gutiérrez et al., 2018).

2. Functional, higher-order, and discrete extensions

A functional extension replaces convex bodies by integrable log-concave functions. For $(\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$0, the functional polar projection body $(\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$1 is defined from the level sets

$(\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$2

and the main inequality takes the form

$(\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$3

If $(\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$4, equality holds if and only if

$(\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$5

for some $(\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$6-dimensional simplex $(\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$7 containing the origin (Alonso-Gutiérrez et al., 2018).

A second functional route uses a Berwald-type monotonicity principle on epigraphs of log-concave functions. For an integrable log-concave $(\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$8 and a concave function $(\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$9 on

$\Pi^*K$0

the map

$\Pi^*K$1

is decreasing for $\Pi^*K$2. 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 $\Pi^*K$3 (Alonso-Gutiérrez et al., 2019).

Higher-order extensions replace the usual covariogram and projection body by $\Pi^*K$4-fold analogues. The resulting higher-order Zhang inequality is

$\Pi^*K$5

with equality if and only if $\Pi^*K$6 is an $\Pi^*K$7-dimensional simplex. The case $\Pi^*K$8 recovers the classical Zhang inequality (Haddad et al., 2023).

The same geometric theme also has a limiting-convolution-body formulation. In that setting, the case $\Pi^*K$9 is described as the exact analogue of Zhang’s inequality, and the sharp equality condition is

$$\frac{2n}{\binom{2n}{n}n^n}\le (\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$$0

for the relevant volume inequality involving $$\frac{2n}{\binom{2n}{n}n^n}\le (\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$$1 (Alonso-Gutiérrez et al., 2013).

A discrete approach replaces Lebesgue measure by lattice-point counting. With the lattice point enumerator

$$\frac{2n}{\binom{2n}{n}n^n}\le (\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$$2

and a mixed measure $$\frac{2n}{\binom{2n}{n}n^n}\le (\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$$3, discrete analogues of the covariogram inequality and the associated projection-body inclusion are proved; by scaling $$\frac{2n}{\binom{2n}{n}n^n}\le (\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$$4 and sending $$\frac{2n}{\binom{2n}{n}n^n}\le (\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$$5, the continuous inequality, and therefore Zhang’s inequality, is recovered (Alonso-Gutiérrez et al., 18 Sep 2025).

3. Affine Sobolev forms and Zhang's energy

A different major usage is Zhang’s affine Sobolev inequality. In one formulation, for an integrable $$\frac{2n}{\binom{2n}{n}n^n}\le (\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$$6-concave function $$\frac{2n}{\binom{2n}{n}n^n}\le (\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$$7,

$$\frac{2n}{\binom{2n}{n}n^n}\le (\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$$8

where $$\frac{2n}{\binom{2n}{n}n^n}\le (\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$$9 is the symmetric decreasing rearrangement and $$\frac{1}{n^n}\binom{2n}{n} \le |K|^{\,n-1}\,|\Pi^\circ K| \le \left(\frac{\kappa_n}{\kappa_{n-1}}\right)^n,$$0 denotes the polar projection body of the function. The same paper proves a stochastic version in expectation for random models built from $$\frac{1}{n^n}\binom{2n}{n} \le |K|^{\,n-1}\,|\Pi^\circ K| \le \left(\frac{\kappa_n}{\kappa_{n-1}}\right)^n,$$1-concave functions and then derives the deterministic generalization

$$\frac{1}{n^n}\binom{2n}{n} \le |K|^{\,n-1}\,|\Pi^\circ K| \le \left(\frac{\kappa_n}{\kappa_{n-1}}\right)^n,$$2

for any rotationally invariant convex measure $$\frac{1}{n^n}\binom{2n}{n} \le |K|^{\,n-1}\,|\Pi^\circ K| \le \left(\frac{\kappa_n}{\kappa_{n-1}}\right)^n,$$3 on $$\frac{1}{n^n}\binom{2n}{n} \le |K|^{\,n-1}\,|\Pi^\circ K| \le \left(\frac{\kappa_n}{\kappa_{n-1}}\right)^n,$$4 (Sola, 4 Sep 2025).

In homogeneous Sobolev spaces, Zhang’s refinement replaces the classical gradient norm by a smaller affine-invariant quantity. The higher-order generalization defines, for $$\frac{1}{n^n}\binom{2n}{n} \le |K|^{\,n-1}\,|\Pi^\circ K| \le \left(\frac{\kappa_n}{\kappa_{n-1}}\right)^n,$$5 and $$\frac{1}{n^n}\binom{2n}{n} \le |K|^{\,n-1}\,|\Pi^\circ K| \le \left(\frac{\kappa_n}{\kappa_{n-1}}\right)^n,$$6 with $$\frac{1}{n^n}\binom{2n}{n} \le |K|^{\,n-1}\,|\Pi^\circ K| \le \left(\frac{\kappa_n}{\kappa_{n-1}}\right)^n,$$7, an affine functional $$\frac{1}{n^n}\binom{2n}{n} \le |K|^{\,n-1}\,|\Pi^\circ K| \le \left(\frac{\kappa_n}{\kappa_{n-1}}\right)^n,$$8 based on directional higher differences or higher directional derivatives, satisfying

$$\frac{1}{n^n}\binom{2n}{n} \le |K|^{\,n-1}\,|\Pi^\circ K| \le \left(\frac{\kappa_n}{\kappa_{n-1}}\right)^n,$$9

and yields the affine embedding

$K$0

For $K$1, this recovers Zhang’s original affine-Sobolev structure (Bullion-Gauthier, 12 Jun 2025).

On $K$2, the relevant object is Zhang’s affine $K$3 energy. For $K$4,

$K$5

with $K$6. This functional is affine invariant,

$K$7

and satisfies

$K$8

It underlies sharp affine Poincaré, Poincaré-Wirtinger, and Poincaré-Sobolev inequalities on $K$9, and the cited paper proves existence of extremals in the subcritical range (Leite et al., 2021).

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$0 Hermitian positive semidefinite matrix $n$1 and a nontrivial permutation $n$2,

$n$3

The refinement is proved in the cited work by a majorization argument, and for positive definite matrices the equality conditions are completely characterized (Lin et al., 2020).

Another usage concerns sectorial matrices. If $n$4 is partitioned as

$n$5

and $n$6, then P. Zhang’s Rotfel’d-type inequality compares $n$7 with block-diagonal terms for concave $n$8. A 2024 refinement introduces a free parameter $n$9 and strengthens Zhang’s theorem by means of the block positivity criterion

$K$0

for sectorial matrices (Fanghong et al., 2024).

The name also appears in generalized matrix-function inequalities. For positive semidefinite matrices $K$1, Zhang et al. proved

$K$2

and a later paper extends this to positive semidefinite block matrices through partial generalized matrix functions $K$3 and $K$4 (Huang et al., 2020).

A further matrix-analytic meaning is Teng Zhang’s hybrid triangle inequality. For $K$5,

$K$6

where

$K$7

The constant $K$8 is stated to be optimal in every dimension $K$9. The 2026 refinement derives operator and eigenvalue versions from a new polar decomposition for the quadratic symmetric modulus (Bourin et al., 28 Jun 2026).

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

$\Pi^*K$0

from Zhang’s theorem on admissible $\Pi^*K$1-tuples. Trudgian used this step with an admissible block of consecutive primes to improve the explicit prime-gap bound from $\Pi^*K$2 million to $\Pi^*K$3 (Trudgian, 2013).

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 $\Pi^*K$4 (Mavraki et al., 8 Dec 2025).

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 $\Pi^*K$5, $\Pi^*K$6, let

$\Pi^*K$7

be the average over $\Pi^*K$8, and let $\Pi^*K$9 be the lower median

$\Pi^\circ K$0

The main theorem is

$\Pi^\circ K$1

and the constant is best possible (Leng, 25 May 2026).

The proof proceeds from the pointwise rearrangement estimate

$\Pi^\circ K$2

where $\Pi^\circ K$3 is the decreasing rearrangement, followed by the classical Hardy inequality. The factor $\Pi^\circ K$4 comes from the half-measure property of the median, and the factor $\Pi^\circ K$5 is the classical best constant in Hardy’s inequality (Leng, 25 May 2026).

The paper also proves the discrete analogue: for $\Pi^\circ K$6, with

$\Pi^\circ K$7

and $\Pi^\circ K$8 the lower median of $\Pi^\circ K$9,

$(\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$00

For $(\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$01, the constant becomes $(\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$02, matching Zhang’s original discrete problem. This places the Olympiad-style question inside a broader $(\mathrm{vol}_n(K))^{n-1}\mathrm{vol}_n(\Pi^*K),$03 Hardy framework and gives a sharp continuous-discrete correspondence (Leng, 25 May 2026).