Domains with mild symmetry

FormalPara Definition

Let D be domain in \({{\mathbb {C}}}^n\) and assume tacitly that all vectors under arrows belong to \({\mathbb {C}}^n\). We say that D is a circular domain, with center \(\, \overrightarrow{0} \, \) if the following holds:

$$\begin{aligned} \overrightarrow{z} \in D, \quad \theta \in {{\mathbb {R}}}\Longrightarrow ( e^{\sqrt{-1}\theta }) \overrightarrow{z} \in D. \end{aligned}$$

D is said to be circular with center \(\overrightarrow{c}\) when \(D-\overrightarrow{c}\) is circular with center \(\overrightarrow{0}\).

Thus the origin centered unit ball \(\{ \vec z \, \Big \vert \Vert \vec z \Vert = 1 \}\) and the standard unit polydisc,

$$\begin{aligned} P \equiv \{ \vec z \,\, \Big \vert \, \, \vert z_i \vert \le 1,\quad i=1 \ldots n \}, \end{aligned}$$

are both circular.

In the convex geometry literature circular bodies are sometimes said to have \(R_\theta \) symmetry. In the functional analysis literature one sometimes speaks of a balanced set.

FormalPara Theorem 1

Let D be a star-shaped origin symmetric domain in \({{\mathbb {C}}}^n\), let \(G_{1,n}\) be the set of 1-dimensional vector subspaces of \({\mathbb {C}}^n\), that is, the (Grassmannian) set of complex lines through \(\overrightarrow{0}\), and let \(d \ell \) be the standard probability measure on \(G_{1,n}\). Then

  1. (1)

    \(\text {vol}(D) \ge \frac{1}{n!} \int _{ \ell \in G_{1,n}} \left( \text {area}(D \cap \ell ) \right) ^n \,d\ell .\)

  2. (2)

    \( \text {vol}(D) = \frac{1}{n!} \int _{ \ell \in G_{1,n}} \left( \text {area}(D \cap \ell ) \right) ^n \,d\ell \text { precisely when } D \text { is circular.}\)

Loosely we say that the volume of D dominates the mean of the nth power of cross-sectional areas of D by lines in \(G_{1,n}\), with equality precisely when D is circular.

This suggests the quest of characterizing all ‘classes’ of bodies by geometric integral inequalities and identities, where classes, geometric, and integral are left to be made more precise (not to mention all). For instance, characterizations of centered bodies in \({\mathbb {R}}^n\) are found in [5, 14]. We seek a mapping (functor?) from classes of bodies to geometric inequalities, perhaps in the spirit of [16], where a parallel is drawn between mean value properties and differential equations. For some results in this direction see [3, 4, 6, 9, 11, 12, 17]. As an intermediate goal, one may aim to characterize the bounded symmetric hermitian domains or Siegel domains in the style and spirit of L. K. Hua’s book [8].

It would also be interesting to provide stability estimates. That is, if the inequality in Theorem 1 is nearly an equality is the body in question nearly circular, perhaps as measured by some kind of modulus of circularity? For extensive discussions of such estimates in convex geometry and geometric inequalities see [7, 15].

FormalPara Proof

Viewing the ambient space \({{\mathbb {C}}}^n\) as \({{\mathbb {R}}}^{2n}\), with its unit sphere \(S^{2n-1}\), we can compute the volume of the region D in polar coordinates. We recall that \( \rho _D ( \omega ) : S^{2n-1} \longrightarrow {\mathbb {R}} \), the radial function of D, is defined by

$$\begin{aligned} \rho _D ( \omega ) = \max \{ t \in [0, + \infty ) \, \vert \, t \omega \in D \}. \end{aligned}$$

With this notation we have

$$\begin{aligned} \text {vol}(D) = \frac{1}{2n} \int _{ \omega \in S^{2n-1}} \left( \rho _D( \omega )\right) ^{2n}\, d \omega , \end{aligned}$$

where \( d \omega \) is the usual surface measure on the sphere \(S^{2n-1}\) induced by the Euclidean metric. Since multiplication by a phase factor stabilizes the sphere and preserves its measure,

$$\begin{aligned} \int _{ \omega \in S^{2n-1}} f(\omega ) \, d \omega = \int _{ \omega \in S^{2n-1}} f( e^{i \theta }\omega ) \, d \omega \end{aligned}$$

for any \(\theta \).

Let \( d \theta \) be the probability measure on the unit circle. Applying Jensen’s inequality [13] to the convex function \(x \mapsto x^n\) on \([0, +\infty )\) we obtain

$$\begin{aligned} \int _{\theta \in S^1} \left\{ \left( \rho _D( e^{ i \theta } \omega )\right) ^{2}\right\} ^n \, d\theta \ge \left\{ \int _{\theta \in S^1} \left( \rho _D( e^{ i \theta } \omega )\right) ^{2}\, d\theta \right\} ^n. \end{aligned}$$

With these observations we have

$$\begin{aligned} \text {vol}(D)= & {} \frac{1}{2n} \int _{ \omega \in S^{2n-1}} \left( \rho _D( \omega )\right) ^{2n} \, d \omega \\= & {} \frac{1}{2n} \int _{ \omega \in S^{2n-1}} \left( \rho _D( e^{ i \theta } \omega )\right) ^{2n} \, d \omega \\= & {} \frac{1}{2n} \int _{ \omega \in S^{2n-1}}\ \int _{\theta \in S^1} \left( \rho _D( e^{ i \theta } \omega )\right) ^{2n} \, d\theta \, d \omega \\= & {} \frac{1}{2n} \int _{ \omega \in S^{2n-1}} \int _{\theta \in S^1} \left\{ \left( \rho _D( e^{ i \theta } \omega )\right) ^{2} \right\} ^n \, d\theta \, d \omega \\\ge & {} \frac{1}{2n} \int _{ \omega \in S^{2n-1}} \left\{ \int _{\theta \in S^1} \left( \rho _D( e^{ i \theta } \omega )\right) ^{2} \, d\theta \right\} ^n \, d \omega \\= & {} \frac{1}{2n} \int _{ \omega \in S^{2n-1}} \left\{ \frac{1}{\pi } area ( D \cap \ell (\omega )) \right\} ^n\, d \omega , \end{aligned}$$

where the inequality is Jensen’s. Notice that equality holds iff \(\rho _D( e^{ i \theta } \omega ) = \rho _D( \omega )\) for almost all (and hence for all) \(\theta \) and \(\omega \) (by continuity), i.e., if and only if D is a circular region.

This nearly completes the proof, but we stated the Theorem in terms of integration over the Grassmannian \(G_{1,n}\) (with probability measure) and we estimated volume by integrating over the sphere \(S^{2n-1}\), the Stiefel manifold if you will (with surface area measure). The two integrals are related by a constant that turns out to be \( 2n ( \pi ^n /n!) \), [14, p. xxi]. Thus

$$\begin{aligned} \text {vol}(D)\ge & {} \frac{1}{2n} \int _{ \omega \in S^{2n-1}} \left\{ \frac{1}{\pi } area ( D \cap \ell (\omega )) \right\} ^n \, d \omega \\= & {} \frac{1}{2n} \cdot 2n \cdot \dfrac{\pi ^n}{n!} \int _{ \ell \in G_{1,n}} \left\{ \frac{1}{\pi } area ( D \cap \ell ) \right\} ^n \, d \omega \\= & {} \dfrac{1}{n!} \int _{ \ell \in G_{1,n}} \left( \text {area}(D \cap \ell ) \right) ^n \,d\ell . \end{aligned}$$

\(\square \)

FormalPara Theorem 2

Let A and B be origin-symmetric star bodies in \({{\mathbb {C}}}^n\) with A circular. If for every complex line \(\ell \) through the origin in \({{\mathbb {C}}}^n\) we have

$$\begin{aligned} \text {area}(A \cap \ell ) \le \text {area} (B \cap \ell ), \end{aligned}$$

then A has smaller or equal volume compared to B.

This is a variation on the classical Busemann–Petty problem [1, 2]. It just involves integration over polar coordinates (adapted to complex geometry) together with Jensen’s inequality. The general Busemann–Petty problem has a vast literature and its solutions are powered by the notion of intersection body, introduced by E. Lutwak in the paper [10], which has played a decisive role in the subject, and for which this result may be viewed as a small manifestation.

FormalPara Proof

By the previous theorem (statement and proof),

$$\begin{aligned} \text {vol}(A)= & {} \frac{1}{n!} \int _{G_{1,n} } \left( \text {area}(A \cap \ell ) \right) ^n \,d\ell \\\le & {} \frac{1}{n!} \int _{G_{1,n}} \left( \text {area}(B \cap \ell ) \right) ^n \,d\ell \\\le & {} \text {vol}(B). \end{aligned}$$

\(\square \)

Note a similar argument allows comparison of bodies K, L in \({\mathbb {R}}^{4n}\) by cross-sections along quaterionic lines (assuming K is quaternion-circular, or \(S^3\)-circular.)