A mathematical preprint presents a multi-output form of the Borell–Brascamp–Leib integral inequality, asking whether each input function can have its own output function. The setup permits the output functions to differ from one input to another. Its main theorem estimates a quantity called K by a λ-weighted power mean of ratios between the outputs’ and inputs’ L1 norms. An L1 norm is the integral-based measure of a function’s size used in this setting.
The theorem is stated for integrable, nonnegative output functions over the full power range −∞ ≤ p ≤ 1/d. The input functions are required to be integrable and to have finite, positive L1 norms. For fixed dimension d, power p and weights λ, the least multiplicative constant is C = 1. The norm ratios are the terms combined by the weighted power mean in the theorem.
The power has its own optimality statement. When m ≥ 2, the power Q_d(p) is optimal: if the multiplicative constant is kept at 1, replacing Q_d(p) with any smaller power makes the inequality fail. The least-constant statement is made for fixed d, p and λ, while the power statement adds the condition m ≥ 2 and keeps C at 1.
Where the power range splits
The derivation from the classical theorem has a narrower range than the main statement. It works for −∞ ≤ p ≤ 1/(d + 1), but fails for 1/(d + 1) < p ≤ 1/d because the integral inequality is outside its valid range. The theorem’s full stated interval nevertheless runs through 1/d.
The proof combines a one-dimensional minimum-type integral inequality with higher-dimensional slice-integral arguments and the composition formula for the power transform.
The theorem sits within a defined mathematical setting. The functions are integrable, each input has a finite, positive L1 norm, and the statement includes weights, dimension and power parameters. For the full stated range, the output functions are nonnegative as well as integrable. These conditions define the class of functions covered by the result.
Special cases and exact examples
Several familiar forms appear as special cases. When the output functions are equal, the formulation recovers the classical multiple-function inequality. At p = 0, it is related to a special case of the multi-output Prékopa–Leindler inequality. Equal outputs provide the common-output recovery, while p = 0 supplies the Prékopa–Leindler comparison.
The p = 0 comparison has a technical boundary. The special formulation uses an ordinary infimum, while the main theorem uses an essential infimum, and the paper says the former does not directly establish the latter. The two infimum conventions are therefore part of how the special case is stated.
The paper constructs equality families when the output functions are proportional, giving cases in which the bound is attained exactly. A change of variables replaces the evaluation point with a general linear combination. The stated optimality results for the multiplicative constant and the power are preserved under that extension.
Those constructions do not provide a complete catalogue of every equality case. The result remains conditional on its assumptions about integrability, finite positive input norms, weights, dimension and powers. The examples establish exact attainment for the families constructed, not a complete equality classification.
On the publication record
The supplied front matter identifies the work as preprint arXiv:2608.23963v1, dated 25 Aug 2026. Its acknowledgment discloses the use of ChatGPT for manuscript organization, translation, language editing and exposition, followed by review and independent verification. The acknowledgment thanks named colleagues for manuscript comments but does not state a funding source.
Paper data and sources
Original title: Borell--Brascamp--Lieb inequality with finitely many output functions
Authors: Takashi Satomi
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
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