Logarithmic Brunn–Minkowski through a sharp variance inequality

Source and verification

This article explains family 091 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Support functions and logarithmic interpolation

The support function $h_K(u)={\sup }_{x\in K}⟨x,u⟩$ determines a convex body. If $0$ is interior to $K$, it is positive on the unit sphere. For a positive function $f$, define the Wulff body

$$W[f]=\bigcap _{u\in {\mathbb{S}}^{n-1}}\{x:⟨x,u⟩\le f(u)\}.$$

A geometric mean of support functions need not itself be a support function. The logarithmic interpolation therefore uses

$$K_{\lambda }=W[h_K^{1-\lambda }{h}_L^{\lambda }],0\le \lambda \le 1.$$

Statement in the manuscript

For origin-symmetric convex bodies $K,L\subset {\mathbb{R}}^n$,

$$|K_{\lambda }|\ge |K{|}^{1-\lambda }|L{|}^{\lambda }.$$

All dimensions are included; smoothness and strict convexity are not assumed. (OpenAI, 2026)

The arithmetic–geometric mean inequality gives $h_K^{1-\lambda }{h}_L^{\lambda }\le (1-\lambda )h_K+\lambda h_L$. Thus $K_{\lambda }\subseteq (1-\lambda )K+\lambda L$. A lower volume bound for this smaller interpolation strengthens the usual logarithmic volume bound for the arithmetic interpolation.

🏷️ Boxes as a normalization check

Let $K={\prod }_j[-a_j,a_j]$ and $L={\prod }_j[-b_j,b_j]$, with positive half-widths. Their Wulff interpolation is the box with half-widths $a_j^{1-\lambda }{b}_j^{\lambda }$. Coordinate directions give containment in that box. Conversely, Hölder’s inequality gives

$$\begin{array}{c}\sum _j|u_j|a_j^{1-\lambda }{b}_j^{\lambda }\\ \le {\left(\sum _j|u_j|a_j\right)}^{1-\lambda }{\left(\sum _j|u_j|b_j\right)}^{\lambda }.\end{array}$$

This support-function inequality puts the box inside $K_{\lambda }$. Its volume is $|K{|}^{1-\lambda }|L{|}^{\lambda }$. The example checks the scaling without classifying every equality case.

🏷️ A smooth partition function

The manuscript approximates a symmetric polytope by soft slab constraints. Fix nonzero vectors $p_i$, widths $h_i>0$, an even integer $q$, and $\epsilon >0$. Set

$$\begin{array}{rl}{\psi }_i(x)& ={\left(\frac{⟨p_i,x⟩}{h_i}\right)}^q,\\ V(x)& =\frac{\epsilon }{2}|x{|}^2+\sum _i{\psi }_i(x),\\ d\mu (x)& =Z^{-1}{e}^{-V(x)}dx.\end{array}$$

Evenness makes the potential convex and symmetric. The Gaussian term ensures integrability even when the slab directions do not span the space.

Let widths vary geometrically, $h_i(t)=h_i(0)e^{t{b}_i}$. Then

$$\begin{array}{rl}{V}^{\prime }& =-q\sum _i{b}_i{\psi }_i,\\ V^{\prime \prime }& =q^2\sum _i{b}_i^2{\psi }_i.\end{array}$$

Differentiating the integral gives the exact identity

$$(\log Z{)}^{\prime \prime }={\mathrm{Var}}_{\mu }(V^{\prime })-{\mathbb{E}}_{\mu }{V}^{\prime \prime }.$$

The source’s central variance assertion is

$${\mathrm{Var}}_{\mu }\left(\sum _i{b}_i{\psi }_i\right)\le \sum _i{b}_i^2{\mathbb{E}}_{\mu }{\psi }_i.$$

It forces $(\log Z{)}^{\prime \prime }\le 0$. The sharp coefficient one is indispensable; an unspecified constant would not give the sign.

🏷️ The dual estimate

The analytic construction uses moment coordinates: a convex potential represents $\mu$ as the image of another log-concave measure under its gradient. A positive matrix field $\tau$ supplies integration-by-parts coefficients. It is bounded above for the regularized problem, but no uniform positive lower bound is assumed.

The manuscript constructs a dense family of centered even functions $u$ as double divergences of matrix fields. With $a_i={\mathbb{E}}_{\mu }{\psi }_i$ and $d_i={\mathbb{E}}_{\mu }(u{\psi }_i)$, its tensor and Bochner identities yield

$$\sum _i\frac{d_i^2}{a_i}\le {\mathbb{E}}_{\mu }{u}^2.$$

Two Cauchy–Schwarz estimates contribute complementary terms. One pairs Hessians with matrix fields; the other uses homogeneity, $D^2{\psi }_{i}x=(q-1)\nabla {\psi }_i$. Their coefficients add to $q$ and cancel the factor $\mathbb{E}\mathrm{tr}(\tau D^2{\psi }_i)=q{a}_i$. (OpenAI, 2026)

The deduction from the dual bound is elementary. Put $F={\sum }_i{b}_i{\psi }_i$ and $u=F-\mathbb{E}F$. Density legitimizes this choice. Then

$$\begin{array}{c}\Vert u{\Vert }_2^2=\sum _i{b}_i{d}_i\\ \le {\left(\sum _i{b}_i^2{a}_i\right)}^{1/2}{\left(\sum _i{d}_i^2/a_i\right)}^{1/2}\\ \le {\left(\sum _i{b}_i^2{a}_i\right)}^{1/2}\Vert u{\Vert }_2.\end{array}$$

If $\Vert u{\Vert }_2=0$, the conclusion is immediate; otherwise divide and square. The research input is the dual estimate and the density theorem, not this final substitution.

🏷️ Limits and consequences

As $q\to \infty$, the soft constraints approach the indicator of the slab intersection away from its measure-zero boundary. Removing Gaussian regularization recovers its volume. Approximation by finite collections of supporting slabs then reaches arbitrary symmetric convex bodies.

The paper also discusses $L_p$ interpolation for $0<p<1$ and a scalar dilation consequence for even log-concave measures through an external transfer theorem. Scalar dilation is distinct from allowing independent coordinate dilations. The central body theorem retains origin symmetry throughout.

🏷️ Key steps and proof outlines

The sharp variance estimate is a covariance domination statement

Let $a_i={\mathbb{E}}_{\mu }{\psi }_i>0$ and let $C_{ij}={\mathrm{Cov}}_{\mu }({\psi }_i,{\psi }_j)$. The central assertion is the matrix inequality

$$C⪯\mathrm{diag}(a_1,\dots ,a_N).$$

Indeed, evaluating both sides on the vector $b=(b_i)$ gives exactly the variance bound in the previous section. This formulation makes its strength clear: all slab perturbations must be controlled simultaneously, including mixed signs of the $b_i$. Separate estimates for $\mathrm{Var}({\psi }_i)$ would not control the off-diagonal covariances.

Differentiating the partition function along one geometric width path inserts precisely that vector $b$. The factor $q^2$ appears both in the variance of $V^{\prime }$ and in $\mathbb{E}{V}^{\prime \prime }$, so coefficient-one domination gives concavity with no leftover error. This is why the proof needs a sharp tensor inequality rather than a dimension-dependent Poincaré estimate. (OpenAI, 2026)

Moment coordinates provide integration by parts adapted to the measure

The source realizes the regularized law as the gradient image of another log-concave law. Its Hessian determines a positive matrix field $\tau$ in the target coordinates. The resulting integration-by-parts formula has the schematic form of a Stein identity: multiplication by a centered coordinate can be replaced by differentiation weighted by $\tau$.

The construction first treats compactly supported approximating laws, obtains uniform upper control from the Gaussian regularization, and then passes to the whole-space limit. The bound $0<\tau \le (4/\epsilon )I$ is pointwise; it does not provide a uniform positive lower eigenvalue. Every use of weighted derivatives and inverse matrices must respect this possible degeneracy at infinity.

For the homogeneous slab functions, Euler’s identity and its differentiated version give

$$x\cdot \nabla {\psi }_i=q{\psi }_i,D^2{\psi }_{i}x=(q-1)\nabla {\psi }_i.$$

Combining the first with the moment-coordinate integration formula yields the normalization $\mathbb{E}\mathrm{tr}(\tau D^2{\psi }_i)=q{a}_i$. These exact factors later cancel the two complementary tensor contributions.

Tensor divergence tests reveal the missing quadratic control

The proof initially restricts the centered even test $u$ to a range of double-divergence operators applied to compactly supported matrix fields. Compact support makes the repeated integrations by parts legitimate. A weighted Bochner identity expresses the resulting quadratic form in derivatives of those fields and in the geometry of the moment coordinates.

The pairing $d_i=\mathbb{E}(u{\psi }_i)$ has two useful representations. One pairs a matrix field with $D^2{\psi }_i$; the other uses the homogeneity relation involving $x$ and $\nabla {\psi }_i$. Weighted Cauchy–Schwarz bounds each representation with complementary coefficients. Their sum is $q$, matching the identity above, and gives ${\sum }_i{d}_i^2/a_i\le \mathbb{E}{u}^2$.

This is a dual estimate: it bounds how strongly any admissible test can correlate with the entire family of slab energies. The final variance deduction is then an application of Hilbert-space duality, as shown earlier. The tensor calculation is needed before that duality argument has the desired sharp constant.

Density eliminates an artificial restriction on the test function

The desired choice $u={\sum }_i{b}_i{\psi }_i-\mathbb{E}{\sum }_i{b}_i{\psi }_i$ need not itself be a compactly supported double divergence. The density section therefore studies the orthogonal complement of the test range. Integration by parts shows that an orthogonal function satisfies the associated weak elliptic equation. Energy estimates and parity reduce that complement to constants. Since the target space consists of centered even functions, the complement there is trivial.

Approximation in $L^2(\mu )$ then extends the dual inequality to the required $u$. Parity is part of this argument, not decoration: origin symmetry of the body is reflected in the even potential and the even test space.

The hard-body limit preserves the direction of concavity

For fixed positive $\epsilon$, increasing even $q$ makes $e^{-{\sum }_i{\psi }_i}$ converge almost everywhere to the indicator of the slab intersection. The Gaussian term is an integrable dominating function, so partition functions converge at each interpolation parameter. Passing to the limit in the multiplicative concavity inequality requires only convergence at its three parameter values. Removing $\epsilon$ recovers volume, and finite supporting-slab approximations recover arbitrary symmetric bodies.

Using the same finite direction list for both endpoint bodies makes the intermediate slab widths geometric means of corresponding support values. Refining that common list converges to the Wulff intersection in the theorem. This last common-direction requirement ensures the limit is the stated logarithmic interpolation, rather than an unrelated interpolation of two polytopes.

See Also

Subgaussian marginals and logarithmic Sobolev inequalities gives another analytic consequence of log-concavity. Mahler volume products separates symmetric and nonsymmetric sharp convex inequalities.

🏷️ Manuscripts and verification scope

• The logarithmic Brunn–Minkowski conjecture — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The scope concerns the symmetric logarithmic Wulff-body volume inequality. The further measure and interpolation consequences are not included in that headline scope. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. The logarithmic Brunn–Minkowski conjecture.