Sharp Reverse Inequality

• Sharp reverse inequalities are precise bounds that invert classical monotonicity and convexity inequalities by providing optimal constants and characterizing unique extremal scenarios.
• They are fundamental tools in areas like convex geometry, probability theory, and operator analysis, with applications ranging from reverse Hölder and Minkowski inequalities to reverse Cheeger and Lieb–Thirring inequalities.
• Recent advances illustrate how phase transitions and rigidity in extremizers, such as symmetric Laplace or one-sided exponential distributions, drive sharper quantitative bounds in complex analysis.

A sharp reverse inequality is a precise, usually extremal, quantitative bound that inverts the direction of a classical monotonicity or convexity-based inequality and gives the optimal constant in the reversed setting. Such inequalities typically characterize the worst-case scenario in function spaces, probability theory, convex geometry, spectral theory, and operator analysis by determining when a reverse comparison holds and by identifying extremizing distributions or objects. Recent advances reveal deep connections to phase-transition phenomena, geometric structure, convex measures, and probabilistic localization methods.

1. Conceptual Framework and Notation

Sharp reverse inequalities arise when standard monotonicity relations—such as Hölder's, Minkowski's, or operator mean inequalities—are reversed on a domain endowed with convexity, symmetry, or concentration structure. Let $f$ be a function, $X$ a random variable or set, and $\|\,\cdot\,\|_p$ denote the $L^p$ or analogous norm.

A prototypical form is: $\|f\|_p \geq C\,\|f\|_q,$ where $p < q$ and $C$ is an explicit, generally optimal constant, typically attained by a distinguished class of extremal objects (e.g., log-concave densities, power distributions, or linear operators under spectral constraint).

Notable domains include:

• Centred log-concave random variables and negative moments (Melbourne et al., 2 May 2025)
• Convex bodies and mixed volumes (Hug et al., 2019)
• Muckenhoupt and $A_\infty$ weights (Hytönen et al., 2012, Parissis et al., 2016, Canto, 2018, Ortiz-Caraballo et al., 2012)
• Operator means in matrix analysis (Furuichi et al., 2018)
• Polynomials in Bombieri–Weyl norm (Etayo, 2019)
• Hardy–Littlewood–Sobolev structures (Ngô et al., 2015)
• Rényi entropy powers (Li, 2017)

2. Core Sharp Reverse Hölder-Type Inequalities

The recent work by Melbourne–Roysdon–Tang–Tkocz establishes a full continuum of sharp reverse Hölder inequalities for centred log-concave random variables (Melbourne et al., 2 May 2025). For $X$ centred and log-concave,

• For $-1 < p \leq 1$:

$\|X\|_p \geq \Gamma(p+1)^{1/p}\cdot \mathbb{E}|X|.$

• For $p \geq 1$:

$$\|X\|_p \leq C_p\cdot \mathbb{E}|X|, \quad C_p = \max\bigl\{ \Gamma(p+1)^{1/p}, \frac{e}{2}\,\|E-1\|_p\bigr\},$$

with $E \sim \operatorname{Exp}(1)$.

There is a unique phase transition at $p_0 \approx 2.9414$:

• For $1 \leq p \leq p_0$, the extremal distribution is symmetric Laplace.
• For $p \geq p_0$, the extremal switches to a shifted one-sided exponential.

Related reverse inequalities include:

• $L_p$–$L_2$ bounds: $\|X\|_p \geq 2^{-1/2} \Gamma(p+1)^{1/p} \|X\|_2$.
• Two-parameter comparison: $$\|X\|_p \geq \Gamma(p+1)^{1/p} \Gamma(q+1)^{1/q} \|X\|_q$$ for $-1 < p \leq 1 \leq q \leq p_0$.

The proof technique is underpinned by Webb’s simplex-slicing bounds and localization/smoothing arguments reducing to mixtures of exponential laws.

3. Reverse Inequalities in Geometric and Spectral Analysis

Reverse Cheeger Inequality

For any planar convex domain $\Omega$ (Parini, 2015): $$J(\Omega) := \frac{\lambda_1(\Omega)}{h_1(\Omega)^2} < \frac{\pi^2}{4},$$ where $\lambda_1$ is the first Dirichlet eigenvalue, and $h_1$ is the Cheeger constant. The extremal sequence is characterized by elongated domains with fixed area, and the optimal constant $\pi^2/4$ is unattainable within the finite-diameter class.

Reverse Minkowski-Type Inequality

For compact convex bodies $K, M \subset \R^n$ (Hug et al., 2019): $V(K, M[n-1]) \leq \frac{1}{n} V_1(K) V_{n-1}(M),$ with equality precisely for $K$ a segment and $M$ a flat body orthogonal to it.

Reverse Bombieri Inequality for Polynomials

For degree-$N$ monic polynomials $P(x) = \prod_{i=1}^N (x - z_i)$ (Etayo, 2019): $$\prod_{i=1}^N \|x - z_i\| \leq \sqrt{N+1} e^{N/2} \|\prod_{i=1}^N (x - z_i)\|.$$ The bound is asymptotically attained for roots equidistributed on the sphere.

Reverse Lieb–Thirring Inequality

For Schrödinger operators on the half-line with self-adjoint boundary (Weder, 1 May 2024): $$\sum_j m_j |\lambda_j| > -\tfrac{1}{4} \bigl( \operatorname{Tr}\! \int_0^\infty V(x) dx + \operatorname{Tr} B \bigr),$$ with $B$ a self-adjoint boundary condition and $V(x)$ a matrix potential; the $1/4$ is optimal.

4. Reverse Inequalities in Convex and Functional Analysis

Reverse Santaló Inequality for the Polarity Transform

For even geometric log-concave functions $f(x) = e^{-\varphi(x)}$ with polarity $\varphi^\circ$ (Artstein-Avidan et al., 2013): $$\int e^{-\varphi}\,dx \int e^{-\varphi^\circ}\,dx \geq a\,c^n |B_2^n|^2,$$ where $c$ is the Bourgain–Milman constant and $a \approx 0.7$ is explicit. Extremizers correspond to convex bodies via indicator functions.

5. Sharp Reverse Properties in Weighted and Operator Theory

Reverse Hölder properties now admit sharp quantitative exponents and constants in weighted settings, notably for:

• $A_\infty$ and $C_p$ weights (Hytönen et al., 2012, Parissis et al., 2016, Canto, 2018):

$$\biggl( \frac{1}{|Q|} \int_Q w^{1+\delta} \biggr)^{1/(1+\delta)} \leq C\, [w]_\ast\, \frac{1}{|Q|} \int_Q w,$$

where $\delta$ and $C$ are explicit as functions of the characteristic, and become optimal as the weight flattens.

• Operator means (Furuichi et al., 2018):

$$A \triangledown_v B \leq \xi (A \#_v B), \qquad A \#_v B \leq \psi (A !_v B),$$

with sharp $\xi,\psi$ arising as extrema of scalar functions over the spectrum range $[s,t]$. These bounds propagate to Tsallis entropy and operator monotone map inequalities.

6. Reverse Inequality Structures in Information Theory and Harmonic Analysis

Reversed Rényi Entropy Power Inequality

For independent random vectors with Rényi entropies (Li, 2017):

• Forward: $N_a(X+Y) \geq N_a(X) + N_a(Y)$ with sharp $a(p)$.
• Reverse: For $p=0$ or $p=2$, $N_{1/2}(X+Y) \leq N_{1/2}(X) + N_{1/2}(Y)$ provided underlying measures are $s$-concave with $s \geq -1/n$.

Connections to convex bodies (intersection, centroid bodies) undergird the conjectured wider validity.

Reverse Young's Convolution Inequality on Hypercube (Beltran et al., 8 Jul 2025)

For $0 < r < 1$, the sharp reverse Young holds: $$\|f * g\|_{\ell^r(\Z^d)} \geq \|f\|_{\ell^{p_r}} \|g\|_{\ell^{p_r}},$$ with $p_r = \frac{2r}{\log_2(2+2^r)}$; the extremal is the indicator of the cube.

7. Phase Transitions, Rigidity, and Extremals

Numerous sharp reverse inequalities reveal a phase transition of extremizers as the parameter crosses a critical value (see log-concave moment case at $p_0$ (Melbourne et al., 2 May 2025)). Rigidity phenomena arise: maximal (or minimal) configurations are unique, and near-attainment of equality forces geometric or probabilistic structure (e.g., spherical suspension space for reverse eigenfunction Hölder (Gunes et al., 2021), minimal energy sets for polynomials (Etayo, 2019), extremal segments for mixed volumes (Hug et al., 2019)). Quantitative stability theorems pinpoint how "almost extremal" implies proximity to the canonical extremizer, often measured in Hausdorff or Gromov–Hausdorff metrics.

Conclusion

Sharp reverse inequalities delineate the extremal landscape of inverse monotonicity in analysis, geometry, and probability. They serve both as powerful classification results (via phase transitions, rigidity, and stability) and as actionable tools for bounding functionals in harmonic analysis, convex geometry, spectral theory, operator algebra, and information theory. Ongoing advances continue to uncover the underlying geometric and probabilistic structures, drive improvements in weighted estimates, and probe the connections between convexity, spectral theory, and entropy with sharp constants and characterizations.