Volume Approximations of Strongly Pseudoconvex Domains

Abstract

In convex geometry, the Blaschke surface area measure on the boundary of a convex domain can be interpreted in terms of the complexity of approximating polyhedra. This approach is formulated in the holomorphic setting to establish an alternate interpretation of Fefferman’s hypersurface measure on boundaries of strongly pseudoconvex domains in $\mathbb {C}^2$. In particular, it is shown that Fefferman’s measure can be recovered from the Bergman kernel of the domain.

Appendix: Power Diagrams in the Heisenberg Group

1.1 The Euclidean Plane

Let $D(a;r)\subset \mathbb {R}^2$ be a disk of radius r centered at $a\in \mathbb {R}^2$. The power of a point $z=(x,y)\in \mathbb {R}^2$ with respect to $D=D(a;r)$ is the number

$$\begin{aligned} {\text {pow}}(z,D)=|z-a|^2-r^2. \end{aligned}$$

Note that if z is outside the disk D, then ${\text {pow}}(z,D)$ is the square of the length of a line segment from z to a point of tangency with $\partial D$. Thus, it is a generalized distance between z and $\partial D$. For a collection $\mathscr {D}$ of disks in the plane, the power diagram or Laguerre–Dirichlet–Voronoi tiling of $\mathscr {D}$ is the collection of all

$$\begin{aligned} {\text {cell}}(D)=\{z\in \mathbb {R}^2:{\text {pow}}(z,D)< {\text {pow}}(z,D^*), \quad \forall D^*\in \mathscr {D}\setminus \{D\}\}, \ D\in \mathscr {D}. \end{aligned}$$

If $\mathscr {D}$ consists of equiradial disks, the power diagram reduces to the Dirichlet–Voronoi diagram of the centers of the disks. In general, the power diagram of any $\mathscr {D}$ gives a convex tiling of the plane (Fig. 2).

Fig. 2

A power diagram in the plane

Power diagrams occur naturally and have found several applications (see [2], for instance). From the point of view of polyhedral approximations, power diagrams (in $\mathbb {R}^{d-1}$) are intimately related to the constant ${\text {ldiv}}_{d-1}$ in (1.2) (see [15] and [7] for explicit details).

1.2 The Heisenberg Group

Let $K(0;\delta )=\{z'\in \mathbb {H}:|z_1|^4+(x_2)^2<\delta ^4\}$ be a Korányi sphere in $\mathbb {H}$ (see (4.1)). We define the horizontal power of a point $z'\in \mathbb {H}$ with respect to $K=K(0;\delta )$ as

$$\begin{aligned} {\text {hpow}}(z',K)= {\left\{ \begin{array}{ll} |z_1|^2-\sqrt{\delta ^4-(x_2)^2}, &{} \text {if}\ |x_2|\le \delta ^2;\\ \infty , &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

Note that $K_c:=K\cap \{x_2=c\}$ is a (possibly empty) disk in the $\{x_2=c\}$ plane, and ${\text {hpow}}((z_1,x_2),K)={\text {pow}}(z_1,K_{x_2})$, where the right-hand side—being a generalized distance—is set as $\infty$ when $K_{x_2}$ is empty. ${\text {hpow}}$ is then extended to all Korányi spheres to be left-invariant under $\cdot _{\scriptscriptstyle \mathbb {H}}$ (defined in Sect. 4). For a collection $\mathscr {K}$ of Korányi spheres in $\mathbb {H}$, define the horizontal power diagram or Laguerre–Korányi tiling of $\mathscr {K}$ to be the collection of all

$$\begin{aligned} {\text {hcell}}(K)=\left\{ z'\in \bigcup _{K\in \mathscr {K}}K:{\text {hpow}}(z',K)<{\text {hpow}}(z',K^*), \quad \forall K^*\in \mathscr {K}\setminus \{K\}\right\} , K\in \mathscr {K}. \end{aligned}$$

Then, ${\text {hcell}}(K)\subset \;K$, for all $K\in \mathscr {K}$ (Fig. 3).

Fig. 3

A $\{x_1=0\}$-slice of a horizontal power diagram in $\mathbb {H}$

We now give two reasons why this concept is useful for us. Let

$$\begin{aligned}&{\text {dil}}_\xi :(z_1,x_2)\mapsto (\xi z_1,\xi ^2 x_2),\\&{\text {dil}}_{w',\xi }:z'\mapsto w'\cdot _{\scriptscriptstyle \mathbb {H}}{\text {dil}}_\xi (-w'\cdot _{\scriptscriptstyle \mathbb {H}}z') \end{aligned}$$

be the dilations in $\mathbb {H}$ centered at the origin and $w'$, respectively. Then,

• (1) ${\text {dil}}_{w',\xi }(K(w',\delta ))=K(w',\xi \delta )$,

• (2) ${\text {hpow}}({\text {dil}}_{w',\xi }(z'),K(w',\delta ))=\xi ^2{\text {hpow}}(z',K(w',\xi ^{-1}\delta ))$, and

• (3) if $\mathscr {K}=\{K_j:=K(a_j,\delta _j):j=1,\ldots ,m\}$, then, ${\text {dil}}_{a_j,\xi }{\text {hcell}}\big (K_l\big ) \cap {\text {dil}}_{a_k,\xi }{\text {hcell}}\big (K_j\big )=\emptyset$, for all $1\le l<j\le m$ and $\xi \le 1$.

Now, consider the Siegel domain $\mathcal {S}$ and the function $f_\mathcal {S}$ studied in Sect. 4. The cuts of any $f_\mathcal {S}$-polyhedron P over $J\subset \partial S$ project to a collection $\mathscr {K}_P$ of Korányi balls in $\mathbb {C}\times \mathbb {R}$ that form a covering of $J'$. The (open) facets of P project to the horizontal power diagram of $\mathscr {K}_P$. This perspective facilitates the proof of

Lemma 6.1

The cuts of $f_{\mathcal {S}_\lambda }$, $\lambda >0$, are Jordan measurable and satisfy the doubling property (3.3) for any $\delta _{f_{\mathcal {S}_\lambda }}>0$ and $D(t)=(1+t)^3$.

Proof

The Jordan measurability of the cuts is obvious. Now, without loss of generality, we may assume $\lambda =1$ (the map $(z,w)\mapsto (\lambda z,\lambda w)$ can be used to handle the other cases). Let $H\subset \partial \mathcal {S}$ be a compact set, $\{w^j\}_{1\le j \le m}\subset H$, $\{\delta _j\}_{1\le j \le m}\subset (0,\infty )$ and $t>0$. For $j=1,\ldots ,m$, let

$$\begin{aligned}&C_j(t):=C(w_j,(1+t)\delta _j;f_\mathcal {S}),&\\&v^j=(w^j)'=(w_1^j,u_2^j),&\end{aligned}$$

and (see (4.1))

$$\begin{aligned} K_j(t):=C_j(t)'=K\left( v^j;\sqrt{(1+t)\delta _j}\right) . \end{aligned}$$

Consider $\mathscr {K}=\{K_j(t):1\le j\le m\}$ and the corresponding horizontal power diagram $\{{\text {hcell}}_j(t)={\text {hcell}}(K_j(t)):1\le j\le m\}$. Then, setting $dz'=dx_1dy_1dx_2$, we have, by a change of variables and (1), (2) and (3) above, that

$$\begin{aligned}&{\text {vol}}\left( \bigcup _{j=1}^{m}C_j(t)\right) \\&\quad =\int _{\cup _{j=1}^m K_j(t)} \max \limits _{1\le j\le m}\left\{ {\text {Re}}{\sqrt{\delta _j^2-(x_2-u_2^j+2{\text {Im}}z_1\overline{w_1}^j)}}-|z_1-w_1^j|^2\right\} dz'\\&\quad =\int _{\cup _{j=1}^m K_j(t)}\max \limits _{1\le j\le m}\{-{\text {hpow}}(z',K_j(t))\}dz'\\&\quad = -\sum \limits _{j=1}^{m}\int _{{\text {hcell}}_j(t)}{\text {hpow}}( z',K_j(t)) dz'\\&\quad =-(1+t)^2\sum \limits _{j=1}^{m}\int _{{\text {dil}}_{v^j,\frac{1}{\sqrt{1+t}}}({\text {hcell}}_j(t))} {\text {hpow}}\left( {\text {dil}}_{v^j,\sqrt{1+t}}(\zeta ),K_j(t)\right) d\zeta \\&\quad =-(1+t)^3\sum \limits _{j=1}^{m}\int _{{\text {dil}}_{v^j,\frac{1}{\sqrt{1+t}}}({\text {hcell}}_j(t))} {\text {hpow}}\left( \zeta ,K_j(0)\right) d\zeta \\&\quad \le (1+t)^3\int _{\cup _{j=1}^m K_j(0)} \max \left\{ -{\text {hpow}}\left( \zeta ,K_j(0)\right) :1\le j\le m\right\} d\zeta \\&\quad = (1+t)^3{\text {vol}}\left( \bigcup \limits _{j=1}^{m}C_j(0)\right) ,\quad \forall t\ge 0. \end{aligned}$$

$\square$

The computations in the above proof also show that

$$\begin{aligned} l_{{\text {kor}}}=\lim _{n\rightarrow \infty }\sqrt{n}\inf \left\{ -\sum \limits _{K\in \mathscr {K}}\int _{{\text {hcell}}(K)}{\text {hpow}}(z',K)dz':I\subset \bigcup _{K\in \mathscr {K}}K,\; \#(\mathscr {K})\le n \right\} , \end{aligned}$$

where I is the unit square in $\mathbb {C}\times \mathbb {R}$ (see Sect. 4). Our proof of Lemma 4.1 yields bounds for $l_{{\text {kor}}}$ as follows:

$$\begin{aligned} 0.0003\approx \frac{4\sqrt{2}}{\pi ^23^7}\le l_{{\text {kor}}}\le \frac{5\sqrt{5}\pi }{3\sqrt{2}}\approx 8.2788. \end{aligned}$$

It would be interesting to know if computations, similar to the ones carried out by Böröczky and Ludwig in [7] for ${\text {ldiv}}_2$, can be done to find the exact value of $l_{{\text {kor}}}$.