Projective transformations of convex bodies and volume product

The volume product of a convex body $K\subset\mathbb{R}^{n}$ whose interior contains the origin is the quantity $\operatorname{vp}(K)\coloneqq|K|\cdot|K^{\circ}|$. Here $|\cdot|$ denotes the standard Lebesgue measure in $\mathbb{R}^{n}$, and $K^{\circ}=\{y\in\mathbb{R}^{n}\mid\langle x,y\rangle=\sum_{i=1}^{n}x_{i}y_{i}\leq 1\,\,\forall x\in K\}$ is the polar body of $K$. While it is well known [Santaló 1949] that the volume product among convex bodies whose barycenter is centered at the origin is maximized only by ellipsoids, the lower bound is still unknown in dimension greater than $2$. More precisely, the Mahler conjecture claims that any convex body $K$ containing the origin in its interior should satisfy the following optimal inequality:

The conjectured lower bound is reached by any $n$-simplex whose vertices span $\mathbb{R}^{n}$ and sum up to zero. Mahler actually proved the inequality in dimension $n=2$ [Mahler 1939], but so far the conjecture remains open for higher dimensions.

If we endow the space of convex bodies in $\mathbb{R}^{n}$ with the usual Hausdorff topology, the volume product becomes a continuous function over the space $\mathcal{K}_{0}^{n}$ of convex bodies in $\mathbb{R}^{n}$ containing the origin in their interior. A standard compactness argument then shows that the functional $\operatorname{vp}:\mathcal{K}_{0}^{n}\to(0,\infty)$ attains its minimum. According to [Santaló 1949] any local minimizer $K$ of the volume product satisfies that its barycenter $\mathbb{G}_{K}$ and the barycenter $\mathbb{G}_{K^{\circ}}$ of its polar coincide with the origin. Now recall that the covariance matrix $\operatorname{cov}(K)$ of a convex body $K$ whose barycenter is centered at the origin is the symmetric matrix with the following entries:

In this paper we propose a short proof of the following result due to Klartag.

It follows from this result that a strong version of the slicing conjecture implies the Mahler conjecture. This strong slicing conjecture states that the homogeneous quantity $\det[\operatorname{cov}(K)]/|K|^{2}$ is maximized among convex bodies with barycenter at the origin by any $n$-simplex whose vertices span $\mathbb{R}^{n}$ and sum up to zero, see [Klartag 2018] for further details.

Klartag’s proof of Theorem 1.1 is based on a variational argument in the space of projective images of $K$ that makes substantial use of the Laplace transform. In this paper we will also use a variational argument in the space of projective images of $K$ in order to recover Theorem 1.1, but our computation will be considerably simpler. More precisely, in section 2 we consider the functional $\mathcal{F}_{K}(x,\xi)\coloneqq\operatorname{vp}\left((K^{\circ}+\xi)^{\circ}+x\right)$ which is well-defined for any convex body $K$ in $\mathcal{K}_{0}^{n}$ on an open neighborhood of the origin in $\mathbb{R}^{n}\times\mathbb{R}^{n}$. By [Santaló 1949] we will easily observe (Proposition 2.1) that the origin is a critical point of the functional $\mathcal{F}_{K}$ if and only if $\mathbb{G}_{K}=\mathbb{G}_{K^{\circ}}=0$. Then an elementary computation (Proposition 2.2) will lead us to the following expression for the Hessian matrix at such a critical point:

Finally we will check (Proposition 2.3) that the above Hessian is positive semi-definite if and only if the symmetric matrix inequality $(n+2)\cdot\operatorname{cov}(K^{\circ})\geq[(n+2)\cdot\operatorname{cov}(K)]^{-1}$ holds true. This implies Theorem 1.1.

Our functional $\mathcal{F}_{K}$ is defined using special projective transformations of the convex body, which are obtained as specific combinations of translations of the polar body with translations of the body itself. In section 3 we will prove that this functional captures all the relevant information about variational properties of the volume product on the space of projective images of a given convex body. More precisely, we will first prove in Proposition 3.1 that once a suitable affine coordinate chart is fixed, the projective transformations close to the identity map transform the convex body $K$ into a new convex body which still contains the origin in its interior. We will then show in Proposition 3.2 that the volume product of these small projective perturbations of our convex body factors through the map $\mathcal{F}_{K}$.

Finally in section 4 we will exhibit an interesting family of convex bodies in dimension $2$ that are critical points of the volume product on the space of their own projective images. More precisely, we will show the following.

The main relevant property of this family is that part of these critical points are saddle points. The last property that there exists small deformations of $\Delta$ verifying (1.1) is also remarkable for the following reason. Recall that the $2$-dimensional Euclidean unit ball $B$ also satisfies $\operatorname{tr}[\operatorname{cov}(B)\operatorname{cov}(B^{\circ})]=1/8$, and that Kuperberg in a private communication to Klartag shows that the Euclidean ball of any dimension is a local maximizer of the functional $K\mapsto\operatorname{tr}[\operatorname{cov}(K)\operatorname{cov}(K^{\circ})]$ in the class of centrally-symmetric convex bodies with sufficiently smooth boundary, see [Klartag 2018, p.78]. At some point it was not clear if this upperbound was a global one, until it was disproved in [Klartag 2018, Proposition 1.5] for large dimensions. Our family discards any non-symmetric analog of Kuperberg’s local result near the $n$-simplex, even in low dimensions.

The purpose of this section is to give a short proof of Theorem 1.1.

Fix a convex body $K$ of $\mathbb{R}^{n}$ containing the origin in its interior, and consider the functional

which is well-defined in an open neighbourhood $U_{K}$ of $(0,0)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$.

Recall the barycenter of a convex body $A$ is defined by the formula $\mathbb{G}_{A}\coloneqq|A|^{-1}\int_{A}xdx$.

In particular the origin is a critical point of the functional $\mathcal{F}_{K}$ if and only if $\mathbb{G}_{K}=\mathbb{G}_{K^{\circ}}=0$.

As observed in the introduction, the first part of this statement implies Theorem 1.1. The second part will be useful in the last section.

The purpose of this section is to show that the volume product on the space of projective images of a given convex body $K$ factors through the map $\mathcal{F}_{K}$. Therefore the functional $\mathcal{F}_{K}$ captures all the relevant information about variational properties of the volume product on the space of projective images of a given convex body.

First recall some material about projective transformations (also called homographies). Consider the real projective space $\mathbb{RP}^{n}=(\mathbb{R}^{n+1}\setminus\{0\})/\mathbb{R}^{*}$. We will make use of the projective coordinates $[x_{0},\ldots,x_{n}]$ for points in $\mathbb{RP}^{n}$. Recall that a projective transformation is a diffeomorphism of $\mathbb{RP}^{n}$ induced by a linear isomorphism of $\mathbb{R}^{n+1}$ as follows: given $A=(a_{ij})\in\operatorname{GL}_{n+1}(\mathbb{R})$ the induced projective transformation $\mathbb{P}{A}$ is

This allows to identify the group of projective transformations $\operatorname{Proj}(\mathbb{RP}^{n})$ with the projective linear group $\operatorname{PGL}_{n}(\mathbb{R})=\operatorname{GL}_{n+1}(\mathbb{R})/(\mathbb{R}^{\ast}\cdot I_{n+1})$ and defines a natural topology on it. Here $I_{n+1}$ denotes the identity matrix. We will use the affine coordinate chart

where $H_{0}=\{[x_{0},x_{1},\ldots,x_{n}]\mid x_{0}=0\}$. Its inverse map is $F^{-1}[x_{0},\ldots,x_{n}]=(x_{1}/x_{0},\ldots,x_{n}/x_{0})$.

Given two nested open balls centered at the origin $B_{1}\varsubsetneq B_{2}$, we define the open subset of $\mathcal{K}_{0}^{n}$

where $\operatorname{cl}$ and $\operatorname{int}$ refer to the closure and the interior respectively.

Now we will reparametrize this open neighbourhood $\mathcal{U}$ in order to show that the volume product of projective images of $K$ factors through the map $\mathcal{F}_{K}$.

Therefore, in order to study the variational properties of the volume product of projective images of a convex body $K\in\mathcal{K}_{0}^{n}$, it is enough to study the associated functional $\mathcal{F}_{K}$.

Fix two numbers $b>0$ and $q\in[-\frac{1}{b},b]$, and set $r=\sqrt{1+q^{2}}$ and $c=\frac{1+bq}{\sqrt{1+q^{2}}}$.

We consider the pentagon $P_{q,b}\subset\mathbb{R}^{2}$ whose vertices are $(-\frac{1}{b},0)$, $(q,\pm r)$ and $(b,\pm c)$. By construction $P=P_{q,b}$ is symmetric with respect to the $x$-axis:

By the previous lemma, if the barycenter of $P$ lies at the origin, the same holds for its dual $-P$, and thus $P$ is a critical point for the volume product on the space of projective images of $P$.

The next lemma gives the exact condition on the pair $(q,b)$ which ensures that $P$ is such a critical point.

As concrete examples of pairs $(q,b)$ such that $P$ is critical, i.e. solutions of the equation in Lemma 4.2, we can consider the following pairs:

• •

  the two pairs $(-1/\sqrt{2},\sqrt{2})$ and $(1/\sqrt{2},1/\sqrt{2})$ at which the pentagon degenerates respectively into an equilateral triangle $\Delta$ and its dual $-\Delta$,

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  the pair $(-\cos(2\pi/5)/\sqrt{\cos(\pi/5)},\sqrt{\cos(\pi/5)})$ at which the pentagon is regular,

• •

  the remarkable pair $(0,1/\sqrt[4]{3})$ which corresponds to a critical saddle point as we will show later.

In addition to these few simple concrete examples, the next lemma 4.3 shows that we have indeed a whole continuous family of critical pentagons $\{P_{q}\}_{q\in[-1/\sqrt{2},1/\sqrt{2}]}$.

As mentioned above, the extremal pentagons $P_{\pm 1/\sqrt{2}}$ degenerate to equilateral triangles.

Now we explain how to compute the Hessian of the functional $\mathcal{F}_{P}$ associated to some pentagon $P$ belonging the above family $\{P_{q}\}$. For this, we must compute the inertia matrix $I\coloneqq I(P)=\int_{P}x_{i}x_{j}dx$ and the area $|P|$ of such a pentagon. Since these quantities are additive with respect to disjoin union, we can perform these computations by decomposing the pentagon into triangles and using the following lemma.

By symmetry, the inertia matrix of such critical pentagon is diagonal so we only need to compute the terms $I_{xx}(P)$ and $I_{yy}(P)$. By decomposing as before $P_{q}$ in six pairewise symmetrical triangles we get first that

Using the same decomposition and Lemma 4.4 we derive that