Wulff bodies and the reduction to a variance bound

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

Let \(K\) and \(L\) be convex bodies in \(\mathbb R^n\) that are symmetric about the origin. In every direction \(u\) take the geometric mean \(h_K(u)^{1-\lambda}h_L(u)^\lambda\) of their support numbers, and intersect the half-spaces \(\langle x,u\rangle\le h_K(u)^{1-\lambda}h_L(u)^\lambda\). The logarithmic Brunn-Minkowski inequality says that the volume of this intersection is at least \(|K|^{1-\lambda}|L|^\lambda\). According to [OpenAI-L] it was conjectured by Böröczky, Lutwak, Yang and Zhang in 2012 and proved by them in the plane. [OpenAI-L] gives a proof in every dimension. Liu, Sun and Xiong extended it to convex bodies whose centroid is the origin, with the cases of equality [LSX]; Section 5 of The variance bound and the inequality states their results.

This lesson states the inequality, proves the facts about support functions that it needs, and reduces it to an inequality between a variance and a mean for explicit smooth probability densities, the variance bound.

The lesson follows [OpenAI-L, Sections 1, 2 and 8.1]. It uses the theorems of dominated and monotone convergence for the Lebesgue integral in \(\mathbb R^n\) (core course Measure and Integration).

1. Support functions and Wulff bodies

A convex body in \(\mathbb R^n\) is a compact convex set with an interior point. Its support function is \[ h_K(u)=\max_{x\in K}\langle x,u\rangle\qquad(u\in\mathbb R^n). \] We use it mostly on the unit sphere \(S^{n-1}\). The volume of \(K\) is written \(|K|\).

Lemma 1.1. Let \(K\) and \(L\) be convex bodies.

  1. \(|h_K(u)-h_K(v)|\le\varrho\,|u-v|\), where \(\varrho\) is the largest norm of a point of \(K\). In particular \(h_K\) is continuous.
  2. \(K\) is the set of the \(x\) with \(\langle x,u\rangle\le h_K(u)\) for all \(u\in S^{n-1}\).
  3. If \(K=-K\), then \(h_K(-u)=h_K(u)\), a ball about \(0\) of some radius \(r>0\) lies in \(K\), and \(h_K\ge r\) on the sphere.
  4. \(h_{aK+bL}=a\,h_K+b\,h_L\) for \(a,b\ge0\) with \(a+b>0\), and \(aK+bL\) is a convex body.

Proof. (1) If \(h_K(u)=\langle x,u\rangle\), then \(h_K(v)\ge\langle x,v\rangle\ge h_K(u)-\varrho|u-v|\); exchange \(u\) and \(v\).

(2) A point of \(K\) satisfies the inequalities by the definition of \(h_K\). Let \(y\notin K\), and let \(x_0\) be a point of \(K\) nearest to \(y\). For \(x\in K\) and \(0<s\le1\) the point \(x_0+s(x-x_0)\) lies in \(K\), so its distance to \(y\) is at least \(|y-x_0|\): \[ -2s\,\langle y-x_0,x-x_0\rangle+s^2|x-x_0|^2\ge0 . \] Divide by \(s\) and let \(s\to0\): \(\langle y-x_0,x-x_0\rangle\le0\). With \(u=(y-x_0)/|y-x_0|\) this says \(\langle x,u\rangle\le\langle x_0,u\rangle\) for all \(x\in K\). So \(h_K(u)=\langle x_0,u\rangle\), and this is \(\langle y,u\rangle-|y-x_0|<\langle y,u\rangle\).

(3) The first statement follows from \(\langle x,-u\rangle=\langle-x,u\rangle\). If a ball of radius \(r\) about a point \(a\) lies in \(K\), then so does the ball of radius \(r\) about \(-a\), and the midpoints of the points of the two balls form the ball of radius \(r\) about \(0\). Then \(h_K(u)\ge\langle ru,u\rangle=r\).

(4) The maximum of \(\langle ax+by,u\rangle\) over \(x\in K\), \(y\in L\) is \(a\,h_K(u)+b\,h_L(u)\). The set \(aK+bL\) is compact and convex, and it contains a translate of a ball. \(\blacksquare\)

Definition. For a positive continuous function \(f\) on \(S^{n-1}\), the Wulff body \(\mathcal W[f]\) of \(f\) is the set of the \(x\in\mathbb R^n\) with \[ \langle x,u\rangle\le f(u)\qquad\text{for all }u\in S^{n-1}. \]

Lemma 1.2. Let \(f,g\) be positive continuous functions on the sphere.

  1. \(\mathcal W[f]\) is a convex body. It contains the ball of radius \(\min f\) about \(0\) and lies in the ball of radius \(\max f\).
  2. If \(f\le g\), then \(\mathcal W[f]\subset\mathcal W[g]\). For \(c>0\), \(\mathcal W[cf]=c\,\mathcal W[f]\).
  3. \(\mathcal W[h_K]=K\) for a convex body \(K\) with \(0\) in its interior.
  4. If \(f\) is even, then \(\mathcal W[f]\) is the set of the \(x\) with \(|\langle x,u\rangle|\le f(u)\) for all \(u\), and \(\mathcal W[f]=-\mathcal W[f]\).
  5. For convex bodies \(K=-K\), \(L=-L\) and \(0\le\lambda\le1\), \[ \mathcal W\bigl[h_K^{1-\lambda}h_L^\lambda\bigr]\subset(1-\lambda)K+\lambda L . \]

Proof. (1) An intersection of closed half-spaces is closed and convex. If \(|x|\le\min f\), then \(\langle x,u\rangle\le|x|\le f(u)\). If \(x\in\mathcal W[f]\) and \(x\ne0\), take \(u=x/|x|\): \(|x|\le f(u)\le\max f\). (2) is clear from the definition. (3) is Lemma 1.1 (2); here \(h_K\) is positive because a ball about \(0\) lies in \(K\). (4) Apply the condition to \(u\) and to \(-u\). (5) For positive numbers, \(a^{1-\lambda}b^\lambda\le(1-\lambda)a+\lambda b\), because the logarithm is concave. So by Lemma 1.1 (4) the function \(h_K^{1-\lambda}h_L^\lambda\) is at most the support function of the convex body \((1-\lambda)K+\lambda L\), and (2) and (3) apply. \(\blacksquare\)

By the definition, the support function of \(\mathcal W[f]\) is at most \(f\). It can be smaller: \(f\) need not be a support function.

2. The inequality and the variance bound

The logarithmic Brunn-Minkowski inequality in dimension \(n\). For all convex bodies \(K=-K\) and \(L=-L\) in \(\mathbb R^n\) and all \(0\le\lambda\le1\), \[ \Bigl|\mathcal W\bigl[h_K^{1-\lambda}h_L^\lambda\bigr]\Bigr|\ \ge\ |K|^{1-\lambda}\,|L|^\lambda . \] This statement is called (LBM) below.

By Lemma 1.2 (5) this is stronger, for symmetric bodies, than the inequality \(|(1-\lambda)K+\lambda L|\ge|K|^{1-\lambda}|L|^\lambda\) of Brunn and Minkowski.

Example 2.1 (boxes). Let \(K\) be the box of the \(x\) with \(|x_i|\le a_i\) and \(L\) the box with \(|x_i|\le b_i\), where \(a_i,b_i>0\). Then \(h_K(u)=\sum_ia_i|u_i|\). Let \(B\) be the box with \(|x_i|\le a_i^{1-\lambda}b_i^\lambda\). The conditions for the coordinate directions show \(\mathcal W[h_K^{1-\lambda}h_L^\lambda]\subset B\). Conversely, for \(x\in B\) and \(0<\lambda<1\), by Hölder's inequality with the exponents \(\tfrac1{1-\lambda}\) and \(\tfrac1\lambda\), \[ \begin{aligned} \langle x,u\rangle&\le\sum_i(a_i|u_i|)^{1-\lambda}(b_i|u_i|)^\lambda\\ &\le h_K(u)^{1-\lambda}\,h_L(u)^\lambda . \end{aligned} \] So the Wulff body is \(B\), and its volume \(\prod_i2a_i^{1-\lambda}b_i^\lambda\) is equal to \(|K|^{1-\lambda}|L|^\lambda\): for boxes with the same axes (LBM) is an equality.

For a probability measure \(\mu\) write \(\mathbb E_\mu f=\int f\,d\mu\) and \(\operatorname{Var}_\mu(f)=\mathbb E_\mu(f-\mathbb E_\mu f)^2\).

The variance bound in dimension \(n\). Let \(N\ge1\), let \(q\ge2\) be an even integer, \(\varepsilon>0\), \(p_1,\dots,p_N\) nonzero vectors of \(\mathbb R^n\) and \(h_1,\dots,h_N>0\). Put \[ \begin{gathered} \psi_i(x)=\Bigl(\frac{\langle p_i,x\rangle}{h_i}\Bigr)^q,\\ V(x)=\frac\varepsilon2|x|^2+\sum_{i=1}^N\psi_i(x),\\ Z=\int_{\mathbb R^n}e^{-V(x)}\,dx,\\ d\mu=Z^{-1}e^{-V(x)}\,dx . \end{gathered} \] Then for all real numbers \(b_1,\dots,b_N\) \[ \operatorname{Var}_\mu\Bigl(\sum_{i=1}^Nb_i\psi_i\Bigr)\ \le\ \sum_{i=1}^Nb_i^2\,\mathbb E_\mu\psi_i . \] This statement is called (VB) below.

The term \(\tfrac\varepsilon2|x|^2\) makes \(Z\) finite, and all polynomials are integrable for \(\mu\). In words: the covariance matrix of the functions \(\psi_i\) is at most the diagonal matrix of their means.

Theorem 2.2. Let \(n\ge2\). If the variance bound holds in dimension \(n\), then the logarithmic Brunn-Minkowski inequality holds in dimension \(n\). In dimension \(1\) the inequality (LBM) holds with equality.

[OpenAI-L, Sections 3 to 7] proves the variance bound in every dimension \(n\ge2\). In this course it is Theorem 4.1 of the lesson The variance bound and the inequality, which rests on two statements from the literature that the course names as hypotheses. Sections 3 to 5 of this lesson prove Theorem 2.2.

3. Soft polytopes

Let \(p_1,\dots,p_N\) be nonzero vectors and \(h_i^0,h_i^1>0\). For real \(t\) let \[ h_i(t)=(h_i^0)^{1-t}(h_i^1)^t , \] and let \(\psi_i(t,\cdot)\), \(V(t,\cdot)\), \(Z(t)\) and \(\mu_t\) be the objects of the variance bound for the widths \(h_i(t)\), with fixed \(q\) and \(\varepsilon\). The density \(e^{-V(t,x)}\) is a smooth replacement of the indicator of the set where \(|\langle p_i,x\rangle|\le h_i(t)\) for all \(i\).

Lemma 3.1. \(Z\) is twice continuously differentiable, and with \(b_i=-q\log(h_i^1/h_i^0)\) \[ \begin{aligned} &\frac{d^2}{dt^2}\log Z(t)\\ &\quad=\operatorname{Var}_{\mu_t}\Bigl(\sum_ib_i\psi_i\Bigr)-\sum_ib_i^2\,\mathbb E_{\mu_t}\psi_i . \end{aligned} \]

Proof. Since \(h_i(t)^{-q}=(h_i^0)^{-q}e^{b_it}\), we have \(\psi_i(t,x)=e^{b_it}\psi_i(0,x)\). So \[ \dot V=\sum_ib_i\psi_i,\qquad\ddot V=\sum_ib_i^2\psi_i , \] where the dots are derivatives in \(t\), and \[ \begin{gathered} \partial_te^{-V}=-\dot V\,e^{-V},\\ \partial_t^2e^{-V}=(\dot V^2-\ddot V)\,e^{-V}. \end{gathered} \] For \(t\) in a bounded interval, \(|\dot V|\), \(\dot V^2\) and \(|\ddot V|\) are at most a polynomial in \(|x|\), and \(e^{-V}\le e^{-\varepsilon|x|^2/2}\). So the two derivatives of \(e^{-V}\) have an integrable bound that does not depend on \(t\), and by dominated convergence, applied to the difference quotients, \(Z\) may be differentiated twice under the integral: \[ \begin{gathered} Z'=-\int\dot V\,e^{-V}dx,\\ Z''=\int(\dot V^2-\ddot V)\,e^{-V}dx . \end{gathered} \] Both are continuous in \(t\), again by dominated convergence. Hence \[ \begin{aligned} (\log Z)''&=\frac{Z''}Z-\Bigl(\frac{Z'}Z\Bigr)^2\\ &=\mathbb E_{\mu_t}\dot V^2-\mathbb E_{\mu_t}\ddot V-(\mathbb E_{\mu_t}\dot V)^2 . \end{aligned} \] \(\blacksquare\)

Corollary 3.2. If the variance bound holds in dimension \(n\), then \(\log Z\) is concave, and for \(0\le\lambda\le1\) \[ Z(\lambda)\ \ge\ Z(0)^{1-\lambda}\,Z(1)^\lambda . \]

Proof. The variance bound for the widths \(h_i(t)\) says that the right side of Lemma 3.1 is at most \(0\). \(\blacksquare\)

4. From soft polytopes to polytopes

Let the vectors \(p_i\) span \(\mathbb R^n\), and let \[ P(t)=\bigl\{x:\ |\langle p_i,x\rangle|\le h_i(t)\text{ for all }i\bigr\}. \] This set is closed and convex, and it is bounded, because \(x\mapsto\max_i|\langle p_i,x\rangle|\) is a norm when the \(p_i\) span.

Lemma 4.1. If the variance bound holds in dimension \(n\), then for \(0\le\lambda\le1\) \[ |P(\lambda)|\ \ge\ |P(0)|^{1-\lambda}\,|P(1)|^\lambda . \]

Proof. Write \(Z_{q,\varepsilon}(t)\) for the integral \(Z(t)\). Fix \(\varepsilon\) and \(t\), and let \(q\to\infty\) through the even integers. If \(|\langle p_i,x\rangle|<h_i(t)\) for all \(i\), every \(\psi_i(t,x)\) tends to \(0\), and \(e^{-V}\) tends to \(e^{-\varepsilon|x|^2/2}\). If \(|\langle p_i,x\rangle|>h_i(t)\) for some \(i\), this \(\psi_i\) tends to infinity and \(e^{-V}\) tends to \(0\). The other points \(x\) lie in the finitely many hyperplanes \(\langle p_i,x\rangle=\pm h_i(t)\), a set of measure \(0\). Since \(e^{-V}\le e^{-\varepsilon|x|^2/2}\), dominated convergence gives \[ Z_{q,\varepsilon}(t)\ \longrightarrow\ Z_\varepsilon(t)=\int_{P(t)}e^{-\varepsilon|x|^2/2}\,dx . \] By Corollary 3.2, \(Z_\varepsilon(\lambda)\ge Z_\varepsilon(0)^{1-\lambda}Z_\varepsilon(1)^\lambda\). Now let \(\varepsilon\) decrease to \(0\). The integrands increase to the indicator of \(P(t)\), and by monotone convergence \(Z_\varepsilon(t)\to|P(t)|\). \(\blacksquare\)

5. All directions

Proof of Theorem 2.2. Let \(n\ge2\), and let \(K=-K\) and \(L=-L\) be convex bodies. For \(0\le t\le1\) let \(f_t=h_K^{1-t}h_L^t\); it is positive, continuous and even, by Lemma 1.1. The sphere has a countable dense subset \(u_1,u_2,\dots\). Let \(D_m\) consist of the coordinate vectors \(e_1,\dots,e_n\) and of \(u_1,\dots,u_m\), and let \(P_m(t)\) be the set of the \(x\) with \[ |\langle p,x\rangle|\le f_t(p)\qquad\text{for all }p\in D_m . \] This is the set \(P(t)\) of Section 4 for the vectors of \(D_m\), which span, with the widths \(h_K(p)\) at \(t=0\) and \(h_L(p)\) at \(t=1\). By Lemma 4.1, \[ |P_m(\lambda)|\ \ge\ |P_m(0)|^{1-\lambda}\,|P_m(1)|^\lambda . \] The sets \(P_m(t)\) decrease as \(m\) grows, and they lie in the bounded set \(P_1(t)\). Their intersection is \(\mathcal W[f_t]\): by Lemma 1.2 (4), \(\mathcal W[f_t]\) lies in every \(P_m(t)\); and a point of all the \(P_m(t)\) satisfies \(|\langle x,u\rangle|\le f_t(u)\) for a dense set of \(u\), hence for all \(u\), because both sides are continuous. So \(|P_m(t)|\) decreases to \(|\mathcal W[f_t]|\), since the measure of a decreasing sequence of sets of finite measure tends to the measure of the intersection. For \(t=0\) and \(t=1\) the intersections are \(\mathcal W[h_K]=K\) and \(\mathcal W[h_L]=L\), by Lemma 1.2 (3). The limit \(m\to\infty\) of the last display is (LBM).

In dimension \(1\) let \(K=[-a,a]\) and \(L=[-b,b]\). The sphere consists of \(\pm1\), \(h_K=a\) and \(h_L=b\), and the Wulff body is the interval of the radius \(a^{1-\lambda}b^\lambda\). Its length is \((2a)^{1-\lambda}(2b)^\lambda\). \(\blacksquare\)

The proof used the variance bound only through Lemma 4.1, for finitely many unit vectors that span.

6. The additive inequality for powers between 0 and 1

For \(0<p<1\) the geometric mean of the support functions is replaced by the mean \(((1-\lambda)h_K^p+\lambda h_L^p)^{1/p}\), which is larger.

Corollary 6.1. Let (LBM) hold in dimension \(n\). Let \(K=-K\), \(L=-L\) be convex bodies in \(\mathbb R^n\), \(0<p<1\) and \(0\le\lambda\le1\), and let \(g=((1-\lambda)h_K^p+\lambda h_L^p)^{1/p}\). Then \[ |\mathcal W[g]|^{p/n}\ \ge\ (1-\lambda)\,|K|^{p/n}+\lambda\,|L|^{p/n}. \]

Proof. For \(\lambda=0\) and \(\lambda=1\) both sides are equal. Let \(0<\lambda<1\). Let \(a=|K|^{1/n}\), \(b=|L|^{1/n}\), \(K_0=a^{-1}K\), \(L_0=b^{-1}L\); these bodies have the volume \(1\), and \(h_K=a\,h_{K_0}\), \(h_L=b\,h_{L_0}\). Let \[ c=\bigl((1-\lambda)a^p+\lambda b^p\bigr)^{1/p},\qquad\theta=\frac{\lambda b^p}{c^p}, \] so that \(0<\theta<1\) and \(1-\theta=(1-\lambda)a^p/c^p\). Then \[ \begin{aligned} g&=c\,\bigl((1-\theta)h_{K_0}^p+\theta h_{L_0}^p\bigr)^{1/p}\\ &\ge c\,h_{K_0}^{1-\theta}\,h_{L_0}^\theta , \end{aligned} \] by the inequality between the weighted arithmetic and geometric means of \(h_{K_0}^p\) and \(h_{L_0}^p\). By Lemma 1.2 (2), \(\mathcal W[g]\) contains \(c\,\mathcal W[h_{K_0}^{1-\theta}h_{L_0}^\theta]\), and by (LBM) the latter has a volume of at least \(c^n\). So \(|\mathcal W[g]|^{p/n}\ge c^p\), and \(c^p\) is the right side. \(\blacksquare\)

7. Exercises

7.1. In the plane let \(K\) be the box with \(|x_1|\le1\), \(|x_2|\le a\) and \(L\) the box with \(|x_1|\le a\), \(|x_2|\le1\), where \(a>0\), \(a\ne1\). Show that \(h_K^{1/2}h_L^{1/2}\) is not the support function of a convex body.

7.2. Let \(n=1\), \(N=1\), \(p_1=1\), and take \(\varepsilon=0\) in the definitions of Section 3, so that \(Z(t)=\int\exp(-(x/h(t))^q)\,dx\). Show that \(\log Z\) is an affine function of \(t\), and deduce that for the probability density proportional to \(e^{-y^q}\) the function \(y^q\) has the mean \(\tfrac1q\) and the variance \(\tfrac1q\). So the variance bound, read for \(\varepsilon=0\), is an equality here.

7.3. Let \(f\) be positive and continuous on the sphere, and \(M\) a convex body with \(0\) in its interior and \(h_M\le f\). Show that \(M\subset\mathcal W[f]\). So \(\mathcal W[f]\) is the largest such body.

7.4. For \(\varepsilon>0\) and a Borel set \(S\) let \(\gamma_\varepsilon(S)=\int_Se^{-\varepsilon|x|^2/2}dx\). Show that the variance bound in dimension \(n\) implies, with the notation of (LBM) and \(f_\lambda=h_K^{1-\lambda}h_L^\lambda\), \[ \gamma_\varepsilon\bigl(\mathcal W[f_\lambda]\bigr)\ \ge\ \gamma_\varepsilon(K)^{1-\lambda}\,\gamma_\varepsilon(L)^{\lambda}. \]

8. Solutions

7.1. Let \(f=h_K^{1/2}h_L^{1/2}\), so \(f(u)^2=(|u_1|+a|u_2|)(a|u_1|+|u_2|)\). By Example 2.1, \(\mathcal W[f]\) is the square with \(|x_i|\le\sqrt a\), whose support function is \(\sqrt a\,(|u_1|+|u_2|)\). At \(u=(1,1)/\sqrt2\) this is \(\sqrt{2a}\), while \(f(u)=(1+a)/\sqrt2\), and \(1+a>2\sqrt a\) for \(a\ne1\). If \(f\) were the support function of a convex body \(M\), then by Lemma 1.1 (2) the body \(M\) would be the set of the \(x\) with \(\langle x,u\rangle\le f(u)\) for all \(u\), which is \(\mathcal W[f]\); but the support function of \(\mathcal W[f]\) is not \(f\).

7.2. The substitution \(x=h(t)y\) gives \(Z(t)=h(t)\int e^{-y^q}dy\), and \(\log h(t)\) is affine in \(t\). The computation of Lemma 3.1 applies for \(\varepsilon=0\), because \(e^{-V}\) decays faster than every power. With \(s=\log(h^1/h^0)\) and \(b=-qs\): \((\log Z)'=-\mathbb E\dot V=qs\,\mathbb E\psi\) must be \(s\), so \(\mathbb E\psi=\tfrac1q\); and \((\log Z)''=b^2(\operatorname{Var}\psi-\mathbb E\psi)\) must be \(0\), so \(\operatorname{Var}\psi=\tfrac1q\). Under \(\mu_t\) the variable \(y=x/h(t)\) has the density proportional to \(e^{-y^q}\), and \(\psi=y^q\).

7.3. By Lemma 1.2 (3) and (2), \(M=\mathcal W[h_M]\subset\mathcal W[f]\).

7.4. In the proof of Lemma 4.1 stop before the limit \(\varepsilon\to0\): \(Z_\varepsilon(\lambda)\ge Z_\varepsilon(0)^{1-\lambda}Z_\varepsilon(1)^\lambda\) for the sets \(P_m(t)\) of Section 5. The measure with the density \(e^{-\varepsilon|x|^2/2}\) is finite, so its values on the decreasing sets \(P_m(t)\) tend to its value on \(\mathcal W[f_t]\).

References