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Borell--Brascamp--Lieb inequality with finitely many output functions

This paper extends the classical Borell--Brascamp--Lieb inequality by establishing a new integral inequality that relates finitely many input functions to distinct output functions, thereby generalizing both the classical multi-function case and specific instances of the multi-output Prékopa--Leindler inequality.

Original authors: Takashi Satomi

Published 2026-08-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Takashi Satomi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of mathematics, there is a branch dedicated to understanding how shapes and spaces behave when they are stretched, squeezed, or combined. Within this field, known as convex geometry, researchers often look for rules that govern the relationship between the size of a whole and the sizes of its parts. Imagine trying to predict the volume of a complex object formed by blending several simpler shapes together. For decades, mathematicians have relied on a powerful set of tools called integral inequalities. These are not just simple arithmetic rules; they are deep statements that allow scientists to compare the total area under a curve, or the total volume of a shape, based on how the individual pieces of that shape relate to one another at every single point. One of the most famous of these rules, known as the Borell–Brascamp–Lieb inequality, acts like a master key. It tells us that if we have a collection of input functions—mathematical descriptions of how a quantity varies across space—and we combine them in a specific way to create a single output function, we can be certain that the total size of that output will be at least as large as a specific combination of the sizes of the inputs. This principle has been fundamental in proving that certain shapes are efficient, that probability distributions behave predictably, and that the geometry of high-dimensional spaces follows strict laws.

For a long time, this master rule had a specific limitation: it required that all the different input functions be funneled into just one single output function. It was as if a rule for mixing ingredients allowed you to pour several different bowls of flour, sugar, and eggs into a single mixing bowl to make one cake, but it offered no guidance if you wanted to make several different cakes at once, each with its own unique recipe. This constraint meant that while the rule was powerful, it could not easily handle situations where multiple distinct outcomes were generated from multiple distinct inputs simultaneously. A new paper by Takashi Satomi removes this restriction. The author has successfully extended the classic rule to a more complex scenario where each input function is paired with its own distinct output function. Instead of forcing everything into a single result, the new inequality allows for a family of outputs, providing a precise mathematical guarantee for the total size of each one based on the sizes of the inputs.

The core of this discovery lies in how the author handles the relationship between the inputs and the outputs. In the traditional version, the rule checked if the combined inputs were always smaller than the single output at every point. Satomi's new version checks a more intricate condition: it looks at the ratio between each specific output and its corresponding input. The rule states that if the weighted average of these ratios, taken across all the different pairs, stays above a certain threshold, then the total sizes of the outputs must follow a specific mathematical relationship with the total sizes of the inputs. This relationship is governed by a special type of average known as a power mean, which can shift its behavior depending on a parameter chosen by the user. The author proves that this new, more flexible rule holds true for a wide range of these parameters, covering everything from simple geometric averages to more complex variations.

What makes this result particularly significant is that it is not just a slight tweak to an old formula; it is a genuine expansion of the mathematical territory. The author demonstrates that when all the output functions happen to be identical, the new rule collapses perfectly back into the old, classic rule, proving that the new work is a true generalization. However, the paper also shows that when the outputs are different, the problem cannot be solved by simply breaking it down into smaller, known problems. The author had to develop a new proof strategy that worked in two stages. First, the rule was proven for a single line of space, using a technique that involved looking at the smallest values of the functions. Then, this one-dimensional result was carefully stacked and combined to build the proof for spaces of any number of dimensions. This method allowed the author to navigate a tricky range of parameters where previous methods had failed, ensuring the rule works for the full spectrum of possibilities.

The paper also addresses the question of whether this new rule is the best it can possibly be. In mathematics, a rule is considered optimal if you cannot make it stronger without breaking it. The author constructs specific examples of functions where the rule holds with perfect equality, meaning the predicted size of the outputs matches the actual size exactly. These examples prove that the constant multiplier in the inequality is as small as it can be, and that the power governing the relationship between the inputs and outputs cannot be changed to a weaker value without the rule failing. This confirms that the new inequality is not just a valid statement, but the sharpest possible version of that statement.

By allowing for multiple distinct outputs, this work opens the door to applying these powerful geometric principles to more complex systems where multiple results are generated in parallel. It connects to other known results in the field, such as the Prékopa–Leindler inequality, showing that this new framework unifies and extends several different mathematical ideas. The author's work stands as a rigorous proof, leaving no room for doubt about the validity of the new inequality. It provides a clearer, more versatile tool for anyone studying how quantities distribute and combine in space, ensuring that the fundamental laws of geometry remain robust even when the systems being studied become more intricate and varied.

Technical Summary: Borell–Brascamp–Lieb Inequality with Finitely Many Output Functions

Problem Statement
The classical Borell–Brascamp–Lieb (BBL) inequality is a fundamental integral inequality in convex geometry and analysis. It relates finitely many input functions $f_1, \dots, f_m$ to a single output function $g$ via weighted power means. Specifically, if the pointwise values of $g$ at a weighted linear combination of inputs dominate the weighted power mean of the input values, then the $L^1$ norm of $g$ is bounded below by a weighted power mean of the $L^1$ norms of the inputs.

This paper addresses a gap in the existing theory: the classical formulation requires a single common output function $g$ for all inputs. The author, Takashi Satomi, investigates whether this inequality can be extended to allow for distinct output functions $g_1, \dots, g_m$ , where each $g_i$ corresponds specifically to the input function $f_i$ . The central question is whether an integral inequality can be established that bounds the essential infimum of the weighted power mean of the ratios $g_i(z_\lambda(x))/f_i(x_i)$ by the weighted power mean of the ratios of their respective $L^1$ norms.

Methodology
The proof of the main theorem proceeds in two primary stages, moving from one dimension to arbitrary dimensions:

One-Dimensional Case (Section 4):
- The author first establishes a minimum-type integral inequality for multiple functions (Lemma 4.3). This is derived by iterating the two-function minimum-type inequality of Brascamp–Lieb (Fact 4.1) via induction on the number of functions.
- This lemma is combined with properties of weighted power means, specifically a reverse Minkowski integral inequality (Lemma 2.9) and a Hölder-type inequality for weighted power means (Lemma 2.7).
- The proof handles the range of powers $-\infty \le p \le 1/d$ by carefully managing the integrability conditions and utilizing the composition formula of the power transform $Q_d$ .
Higher-Dimensional Case (Section 5):
- The general case is deduced from the one-dimensional result using slice integrals and induction on the dimension $d$ .
- The space $\mathbb{R}^{d+e}$ is decomposed into $\mathbb{R}^d \times \mathbb{R}^e$ . The author applies the lower-dimensional results successively to the slice integrals of the functions.
- A crucial component is the dimension-additivity lemma (Lemma 5.1), which relies on the composition formula $Q_e(Q_d(p)) = Q_{d+e}(p)$ for the power transform. This allows the powers arising in the lower-dimensional steps to be combined correctly to yield the final exponent for the higher-dimensional case.

Key Contributions and Results

Main Theorem (Theorem 1.3): The paper establishes the multi-output Borell–Brascamp–Lieb inequality. For integrable functions $f_i, g_i: \mathbb{R}^d \to \mathbb{R}_{\ge 0}$ and a weight $\lambda$ , if $L(x) = M_\lambda^p(g_1(z_\lambda(x))/f_1(x_1), \dots, g_m(z_\lambda(x))/f_m(x_m))$ has an essential infimum $K$ , then:
$K \le M_\lambda^{Q_d(p)} \left( \frac{\|g_1\|_1}{\|f_1\|_1}, \dots, \frac{\|g_m\|_1}{\|f_m\|_1} \right)$
where $Q_d(p) = p/(1-dp)$ is the power transform.
Recovery of Classical Results:
- When all output functions are identical ( $g_1 = \dots = g_m = g$ ), the theorem reduces exactly to the classical BBL inequality for multiple functions (Theorem 1.2).
- When $p=0$ and the hypothesis is imposed pointwise (rather than essentially), the result coincides with a special case of the multi-output Prékopa–Leindler inequality established by Cordero-Erausquin and Maurey.
Optimality (Section 6):
- Multiplicative Constant: The author constructs explicit families of functions (using $\alpha$ -concave functions and indicator functions of convex sets) where equality holds. These examples prove that the multiplicative constant in the inequality is optimal and equals 1.
- Power Exponent: For $m \ge 2$ , the paper demonstrates that the power $Q_d(p)$ on the right-hand side cannot be replaced by any smaller power while maintaining a multiplicative constant of 1. This confirms the sharpness of the exponent.
Extension to General Linear Coefficients (Section 7):
- By applying a change of variables, the author extends the result to cases where the evaluation point is a general linear combination $z_\beta(x) = \sum \beta_i x_i$ rather than a convex combination. The inequality holds with adjusted scaling factors involving the coefficients $\beta_i$ .

Significance and Claims
The paper claims to provide a natural and sharp extension of the Borell–Brascamp–Lieb inequality to settings involving distinct output functions. The significance lies in:

Generalization: It removes the restriction of a common output function, allowing for more flexible integral inequalities involving multiple distinct pairs of input-output functions.
Completeness: Unlike previous derivations that only worked for low-power ranges ( $p \le 1/(d+1)$ ) by combining Hölder and reverse Minkowski inequalities, this work provides a proof valid for the full range $-\infty \le p \le 1/d$ . The author notes that the standard reduction to the classical BBL inequality fails in the high-power range ( $1/(d+1) < p \le 1/d$ ), necessitating the independent proof strategy involving minimum-type inequalities and slice integrals.
Sharpness: The paper rigorously establishes that both the constant and the power exponent in the new inequality are optimal, preventing any further improvement without altering the structure of the inequality.

The work situates itself within the broader context of convex geometry and functional inequalities, connecting the BBL inequality, the Prékopa–Leindler inequality, and the Brunn–Minkowski inequality, while clarifying the relationship between the multi-output results of Cordero-Erausquin–Maurey and the classical BBL framework.

Technical Summary: Borell–Brascamp–Lieb Inequality with Finitely Many Output Functions

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