A geometric-mean law for symmetric convex bodies

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Family 091 · Convex and metric geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-14

1. The question

The Brunn–Minkowski inequality says that combining two shapes by adding their points produces a body with a substantial volume. It is one of the main bridges between convex geometry and inequalities in analysis.

A stronger proposal applies to shapes symmetric about the origin. Instead of taking arithmetic means of their directional support bounds, take geometric means. This generally produces a smaller body. Must its volume still be at least the geometric mean of the original volumes?

That is the logarithmic Brunn–Minkowski conjecture. The manuscript in this family claims it for arbitrary origin-symmetric convex bodies in every dimension. It also derives intermediate power-mean inequalities and, through an earlier transfer theorem, a logarithmic concavity statement for the measure of exponentially dilated symmetric bodies.

The exact definition of the interpolated body matters. A geometric mean of support functions need not itself be a support function; the construction uses an intersection of half-spaces to obtain a genuine convex body.

2. Background and essential definitions

For a convex body $K\subset\mathbb R^n$, the support function is

$$h_K(u)=\max_{x\in K}\langle x,u\rangle,\qquad |u|=1.$$

It records the supporting hyperplane in direction $u$. If $K=-K$, the origin lies in its interior and $h_K$ is positive and even.

Given any positive continuous function $f$ on the unit sphere, its Wulff body is

$$W[f]=\bigcap_{|u|=1}\{x:\langle x,u\rangle\leq f(u)\}.$$

The function supplies directional upper bounds; some may be redundant. The actual support function of the intersection can therefore be smaller than $f$.

For $0\leq\lambda\leq1$, the logarithmic combination of $K$ and $L$ is

$$D_\lambda=W[h_K^{1-\lambda}h_L^\lambda].$$

The ordinary Minkowski combination $(1-\lambda)K+\lambda L$ has arithmetic-mean support function. Arithmetic–geometric mean comparison gives $D_\lambda\subseteq(1-\lambda)K+\lambda L$, explaining why a volume bound for $D_\lambda$ is stronger.

A nonnegative function $F$ is log-concave when

$$F((1-\lambda)s+\lambda t)\geq F(s)^{1-\lambda}F(t)^\lambda.$$

For a positive function, this means $\log F$ is concave. A density $e^{-V(x)}$ is log-concave when $V$ is convex, and even when $V(-x)=V(x)$. Gaussian densities and uniform densities on symmetric convex bodies are familiar examples.

3. Development history and previous partial results

Firey introduced power means of convex bodies, and Lutwak developed the broader $L^p$ Brunn–Minkowski theory. The logarithmic combination appears as the limit of the power mean when $p$ decreases to zero.

Böröczky, Lutwak, Yang and Zhang formulated the symmetric logarithmic conjecture and proved its planar case in The log-Brunn–Minkowski inequality (2012). Their work also related the volume formulation to a logarithmic mixed-volume inequality. The logarithmic Minkowski problem, concerning which measures arise from boundary cone volumes, is related but is not the same theorem.

Saroglou proved the inequality for bodies that are unconditional in a common coordinate basis: each body is unchanged by independently reversing coordinate signs. His 2015 paper connects this geometry to coordinatewise products and to a special diagonal-dilation property. Later results treated common reflection symmetries and unit balls of complex norms.

A second line studied infinitesimal changes of volume. Kolesnikov and Milman formulated local inequalities as spectral estimates for a geometric differential operator and obtained results for $p$ close to one. Putterman established equivalence between local and global formulations in the relevant range. Van Handel proved a local logarithmic result for zonoids, which are limits of sums of segments. Additional work treated bodies of revolution and other symmetry classes.

The manuscript also acknowledges Stancu’s 2018 announcement of an all-dimensional logarithmic Minkowski inequality and its equivalent Brunn–Minkowski conclusion, with flow-asymptotic details deferred there. This digest preserves that attribution rather than presenting the new manuscript as the only prior claim of the general result.

The transfer from volume to even log-concave measures is due to Saroglou’s More on logarithmic sums of convex bodies. The new volume theorem is the input used to activate that earlier implication in every dimension.

4. What the new manuscripts claim

M1 states that for all origin-symmetric convex bodies $K,L\subset\mathbb R^n$, every $n\geq1$, and every $\lambda\in[0,1]$,

$$|W[h_K^{1-\lambda}h_L^\lambda]| \geq |K|^{1-\lambda}|L|^\lambda.$$

No smoothness, common reflection basis or proximity assumption is imposed.

For every $0<p<1$, it deduces the symmetric power-mean inequality

$$\left|W\left[((1-\lambda)h_K^p+\lambda h_L^p)^{1/p}\right]\right|^{p/n} \geq(1-\lambda)|K|^{p/n}+\lambda|L|^{p/n}.$$

This additive volume-power form uses normalization and a reweighting of the interpolation parameter, beyond the elementary comparison of power means.

The measure consequence applies to every even log-concave Radon measure $\mu$. A Radon measure is finite on compact sets, though its total mass may be infinite. Log-concavity here means the corresponding multiplicative inequality under Minkowski combinations of compact sets. The claim is

$$\mu(D_\lambda)\geq\mu(K)^{1-\lambda}\mu(L)^\lambda.$$

The manuscript includes measures supported on a proper linear subspace, not only full-dimensional densities. No probability normalization is required.

Taking $K$ and $L$ to be scalar dilates of the same body gives the scalar (B)-conjecture:

$$t\longmapsto\mu(e^tK)\quad\text{is log-concave on }\mathbb R.$$

The exponential parameter is important. The statement concerns multiplying every coordinate by the same factor $e^t$, not arbitrary independent coordinate dilations for every measure.

5. Why the advance matters

Importance: 5/5. The logarithmic endpoint is a central strengthening of classical convex-volume theory. A theorem for all symmetric bodies would unify numerous special-symmetry and local results, and yield an entire interval of $L^p$ inequalities.

The measure consequence makes the result relevant beyond Euclidean volume. It controls how Gaussian mass, uniform convex-body mass and more general even log-concave mass enter a symmetrically expanding region. Concavity of the logarithm imposes a consistent multiplicative interpolation law between different scales.

The symmetry assumption is analytically meaningful. In local spectral formulations, translations correspond to odd modes. Restricting to even data removes these modes and changes the relevant variance or eigenvalue problem. Symmetry is therefore part of the mechanism, not merely a convenient normalization.

The score concerns the claimed general inequality and its consequences. It does not assert a complete equality classification, quantitative stability theorem, or an unrestricted nonsymmetric analogue. Those are distinct mathematical questions.

6. The main idea

Approximate a body by smooth slab penalties

A symmetric polytope can be written as the intersection of finitely many slabs $|\langle p_i,x\rangle|\leq h_i$. The proof replaces its indicator by a smooth density proportional to

$$\exp\left(-\frac\varepsilon2|x|^2 -\sum_i\left(\frac{\langle p_i,x\rangle}{h_i}\right)^q\right),$$

where $q$ is even and $\varepsilon>0$. Large $q$ penalizes points outside the slabs sharply. Letting $q$ tend to infinity and then removing the Gaussian factor recovers the polytope’s volume.

Geometrically varying the slab widths becomes ordinary linear variation of their logarithms. Differentiating the logarithm of the integral twice produces a variance term minus an expected second derivative. The desired concavity is therefore reduced to a particular variance inequality for sums of even powers of linear forms.

Choose coordinates adapted to moments

Ordinary variance bounds for log-concave measures are not automatically strong enough for this endpoint. M1 instead represents the target measure as the image of a measure with density $e^{-\varphi}$ under the gradient map $x=\nabla\varphi$. These are moment coordinates, arising from earlier moment-measure theory.

The Hessian of $\varphi$, the matrix of second derivatives, supplies a geometry adapted to that map. The manuscript constructs it with a global upper bound and strict positivity locally. It does not assume a uniform positive lower bound everywhere.

A differential operator in these coordinates generates a dense collection of centered even test functions. The possible obstructions are affine functions: subtracting the mean removes constants, while evenness removes linear functions. This is the point where the symmetry enters the functional argument explicitly.

Turn the remainder into squares

For those test functions, a tensor calculation expresses the relevant remainder as a sum of squares. Nonnegativity of these squares yields control of each homogeneous summand separately. Two Cauchy–Schwarz steps then produce the required variance estimate.

The analytic inequality gives concavity for the smooth slab integrals. Approximation passes to polytopes and then arbitrary symmetric bodies. The measure statement follows through Saroglou’s transfer theorem, with a separate reduction to the subspace supporting a lower-dimensional measure.

This chain keeps the genuinely new claimed analytic estimate separate from the established geometric-to-measure implication.

7. Scope, limitations, and open questions

Both bodies must be symmetric about the same origin, and the support bounds must be assembled by the Wulff intersection. Treating their geometric mean as an already-valid support function would omit a necessary part of the definition.

The stated power range is $0<p<1$, with the logarithmic case interpreted separately as the endpoint. The scalar (B)-conclusion uses even log-concave measures and symmetric convex bodies; it is not a claim for arbitrary densities or sets.

The paper discusses a diagonal-dilation property for uniform measures on cubes in an equivalence used for context. That specialized property should not be confused with a general diagonal (B)-theorem for every even log-concave measure. The published consequence summarized here is scalar dilation.

Equality and stability require additional analysis beyond the displayed inequality. The digest reports the proof strategy and dependencies as stated, without verifying the tensor estimate or resolving the relationship between different announced approaches.

8. Sources and further reading

M1’s first section states the volume and power-mean claims; its final section gives the measure theorem and supporting-subspace reduction. The intermediate sections develop moment coordinates, density of even tests and the tensor variance estimate.

Saroglou’s two linked papers distinguish unconditional-body results from the later transfer theorem. Van Handel’s local zonoid paper illustrates an important partial solution. The 2012 paper of Böröczky–Lutwak–Yang–Zhang supplies the conjectural and planar starting point, with full bibliographic details in M1.

Source manuscripts

• M1. OpenAI, The logarithmic Brunn–Minkowski conjecture (September 23, 2026).

Sources consulted on 14 October 2026 at repository revision 1581d4c. Download the manuscript citations.