1 · The essentials
This family of results concerns how the size of a shape changes when shapes are combined or enlarged. The shapes must have no inward dents and must look unchanged when every point is reflected through the origin. The main manuscript establishes the logarithmic Brunn–Minkowski inequality, a particular lower bound on the size of a combination of two such shapes. Its statement applies in every dimension. It also gives a related family of bounds for parameters between zero and one. Further consequences concern ways of measuring size other than ordinary volume, under additional symmetry and concavity requirements. The main result has a Lean formalization, but the supplied description does not specify whether that formalization covers all the consequences.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine two centrally balanced solid shapes and a rule for making an intermediate shape from them. The inequality gives a minimum-size guarantee for the result; the image is incomplete because the mathematical combination is a specific logarithmic one, not simply joining the solids or mixing their material.
Why it matters
The result addresses the logarithmic Brunn–Minkowski conjecture for origin-symmetric convex bodies without restricting the dimension. The manuscript also links that result to a family of inequalities and, through a transfer theorem and a reduction, to inequalities for even log-concave measures. Thus the stated consequences concern both combining shapes and dilating a single shape.
2 · Going further
The precise problem identified in the source is the logarithmic Brunn–Minkowski conjecture for arbitrary origin-symmetric convex bodies in every dimension. Here convexity excludes inward dents, while origin symmetry requires that belongs to the body exactly when does. The main manuscript states the logarithmic inequality in this setting and is identified as Lean-formalized. The supplied abstract does not give the formula for the logarithmic combination or the steps establishing its volume bound.
The same theorem also gives the symmetric Brunn–Minkowski inequality for every . The repository summary specifies an additive Brunn–Minkowski inequality for Lebesgue volume on full-dimensional origin-symmetric convex bodies throughout this range. These restrictions matter: the stated additive result is not a claim about arbitrary bodies or arbitrary parameter values.
For the measure-theoretic consequences, the abstract describes a route rather than the internal argument for the main theorem. Combining the logarithmic result with Saroglou's transfer theorem and a support-subspace reduction yields the logarithmic inequality for every even log-concave Radon measure, together with the scalar-dilation -conjecture. Only one manuscript is supplied, so no relationship with companion manuscripts can be documented from this source.
Useful notions
- Convex bodies and origin symmetry
- Volume in arbitrary dimension
- Logarithmic and combinations
- Log-concave measures
- Scalar dilation
What was known before / what is new
Before
The source identifies the central statement as the logarithmic Brunn–Minkowski conjecture. It names Saroglou's transfer theorem as an ingredient used to obtain the measure-theoretic consequences. It supplies no account of earlier partial results or the historical development of these questions.
New
The main manuscript establishes the logarithmic Brunn–Minkowski inequality for arbitrary origin-symmetric convex bodies in every dimension, and the main result has a Lean formalization. The manuscript also shows the symmetric inequality for and states the logarithmic measure inequality and scalar-dilation -conjecture as consequences. The repository summary explicitly includes the additive result for Lebesgue volume on full-dimensional origin-symmetric convex bodies.
Limits and assumptions
Origin symmetry and convexity are hypotheses of the main geometric result. The additive volume result is stated for full-dimensional bodies and only for ; the measure consequences require even log-concave Radon measures. The statement is specifically about scalar dilation. The main result has a Lean formalization, but the supplied source does not delineate its coverage of the transfer argument or the other consequences. It does not identify remaining open cases, so no broader resolution is asserted here.
Mini glossary
- Convex body —
- A compact convex set with nonempty interior, where convexity means that the segment between any two points of the set stays inside it.
- Origin symmetry —
- The property that reflecting every point through the origin leaves the set unchanged.
- Lebesgue volume —
- The usual notion of volume in the ambient Euclidean space.
- Even measure —
- A measure that assigns the same size to a set and its reflection through the origin.
- Scalar dilation —
- An enlargement or contraction obtained by multiplying every point by the same positive factor.
Keywords : Logarithmic Brunn–Minkowski inequality · $L_p$ Brunn–Minkowski inequality · B-conjecture
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves the logarithmic Brunn–Minkowski inequality for origin-symmetric convex bodies in every dimension, and the scalar-dilation B-conjecture for all even log-concave Radon measures. For Lebesgue volume it also proves the additive Lp Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout .
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
The logarithmic Brunn–Minkowski conjecture
23/09/2026We prove the logarithmic Brunn–Minkowski conjecture for arbitrary origin-symmetric convex bodies in every dimension. The theorem also gives the symmetric Lp Brunn–Minkowski inequality for every . Combined with Saroglou's transfer theorem and a support-subspace reduction, it yields the logarithmic inequality for every even log-concave Radon measure and the scalar-dilation -conjecture.