2. Preliminaries

The standard scalar product of ${\mathbb R}^d$ is denoted by $\langle\cdot\,,\cdot\rangle$ , and the induced norm by $\|\cdot\|$ . The unit ball of ${\mathbb R}^d$ is $B^d$ , and the unit sphere is ${{\mathbb S}^{d-1}}$ . Lebesgue measure on ${{\mathbb R}^d}$ is denoted by $\lambda_d$ .

Hyperplanes and closed half-spaces of ${\mathbb R}^d$ are written in the form

\begin{equation*} H(u,\tau) = \{x\in{\mathbb R}^d \colon \langle x,u\rangle=\tau\},\quad H^-(u,\tau) = \{x\in{\mathbb R}^d \colon \langle x,u\rangle\le\tau\}, \end{equation*}

respectively, with $u\in{{\mathbb S}^{d-1}}$ and $\tau\in{\mathbb R}$ . Let ${\mathcal{H}}$ be the space of hyperplanes in ${\mathbb R}^d$ with its usual topology. For a subset $M\subset{\mathbb R}^d$ , we write

\begin{equation*} {\mathcal{H}}_M\colon= \{H\in{\mathcal{H}}\colon H\cap M\not=\emptyset\}.\end{equation*}

We let ${{\mathcal K}}_d$ denote the space of d-dimensional convex bodies (compact, convex sets with interior points) in ${\mathbb R}^d$ . As usual, it is equipped with the Hausdorff metric. For $K\in{{\mathcal K}}_d$ , let $R_o(K)$ be the radius of the smallest ball with center at the origin o of ${\mathbb R}^d$ that contains K.

For our notation concerning point processes, we refer to [Reference Schneider and Weil27, Sections 3.1 and 3.2]. In particular, given a locally compact topological space E, we let $({\sf N}_s(E), {\mathcal N}_s(E))$ denote the measurable space of simple, locally finite counting measures on E. We often identify a simple counting measure $\eta\in {\sf N}_s(E)$ with its support, using $\eta(\{x\})=1$ and $x\in\eta$ synonymously. A (simple) point process in E is a mapping $X \colon (\Omega,{A},{\mathbb P})\to ({N}_s(E), {\mathcal N}_s(E))$ , where $(\Omega,{A},{\mathbb P})$ is some probability space, such that $\{X(C)=0\}$ is measurable for all compact sets $C\subset E$ . We let $\Theta={{\mathbb E}\,} X$ denote the intensity measure of X. The point process X is a Poisson process if

$${\mathbb P}(X(A)=k)={\textrm{e}}^{-\Theta(A)} \dfrac{\Theta(A)^k}{k!}$$

for $k\in {\mathbb N}_0$ and each Borel set $A\subset E$ with $\Theta(A)<\infty$ . For the independence properties of (simple) Poisson processes, we refer to [Reference Schneider and Weil27, Theorem 3.2.2]. A stationary Poisson hyperplane process in ${\mathbb R}^d$ is a Poisson process X in the space ${\mathcal{H}}$ of hyperplanes whose intensity measure (and hence whose distribution) is invariant under translations. The intensity measure of such a process, assumed to be non-zero, is of the form

$$\Theta(A) =\gamma \int_{{{\mathbb S}^{d-1}}}\int_{-\infty}^\infty {\bf 1}_A(H(u,\tau))\,{\textrm{d}}\tau\,\varphi({\textrm{d}} u)$$

for Borel sets $A\subset {\mathcal{H}}$ . Here $\gamma>0$ is the intensity of X and $\varphi$ is an even probability measure on the sphere ${{\mathbb S}^{d-1}}$ , the spherical directional distribution of X.

We assume now that X is a stationary Poisson hyperplane process in ${\mathbb R}^d$ of intensity $\gamma>0$ and with a non-degenerate spherical directional distribution $\varphi$ . Here, ‘non-degenerate’ means that $\varphi$ is not concentrated on any great subsphere.

For $K\in{{\mathcal K}}_d$ , we let $Z_K$ denote the K-cell defined by X and K, as above. This is a random polytope, since it is almost surely bounded. More precisely, we show the following estimate, which later on, when the intensity tends to infinity, will allow us to restrict ourselves to K-cells contained in a sufficiently large fixed ball.

Lemma 1. Let $K\in{{\mathcal K}}_d$ . There are constants $a,b>0$ , depending only on $\varphi$ , such that

$${\mathbb P}(R_o(Z_K) > b(R_o(K)+x)) \le 2d \,{\textrm{e}}^{-a\gamma x}\quad\mbox{for } x\ge 0.$$

Proof. Since ${\textrm{supp}}\,\varphi$ , the support of the even measure $\varphi$ , is not contained in a great subsphere, we can choose vectors $\pm e_1,\dots,\pm e_d\in {\textrm{supp}}\,\varphi$ positively spanning ${\mathbb R}^d$ . In the following, we write $e_{d+i}\colon=-e_i$ for $i=1,\dots,d$ . We can choose a sufficiently large constant b and sufficiently small, pairwise disjoint neighborhoods $U_i\subset {{\mathbb S}^{d-1}}$ of $e_i$ , $i=1,\dots,2d$ , such that each intersection

$$\label{2.2} P\colon= \bigcap_{i=1}^{2d} H^-(u_i,1) \quad \mbox{with }u_i\in U_i,\,i=1,\dots,2d, $$

is a polytope with $R_o(P)\le b$ . Let $x\ge 0$ . If the numbers $\tau_i$ are such that $R_o(K)\le \tau_i\le R_o(K)+x$ and if $u_i\in U_i$ for $i=1,\dots,2d$ , then

$$R_o\bigg(\bigcap_{i=1}^{2d} H^-(u_i,\tau_i)\bigg) \le R_o((R_o(K)+x)P)=(R_o(K)+x)R_o(P)\le b(R_o(K)+x).$$

The sets of hyperplanes

$$A_i(x)\colon= \{H(u,\tau) \colon u\in U_i,\, R_o(K)\le \tau\le R_o(K)+x\},\quad i=1,\dots,2d,$$

are pairwise disjoint. If $X(A_i(x)) >0$ for $i=1,\dots,2d$ , then $R_o(Z_K)\le b(R_o(K)+x)$ . Therefore, observing that $\Theta(A_i(x)) = \gamma x\varphi(U_i)$ and choosing $0<a\le\varphi(U_i)$ for $i=1,\dots,2d$ , we get

\begin{align*} &{\mathbb P}(R_o(Z_K) > b(R_o(K)+x))\\ &\qquad \le {\mathbb P}( X(A_i(x))=0\mbox{ for at least one } i\in \{1,\dots,2d\})\\ &\qquad = 1-\prod_{i=1}^{2d} (1-{\mathbb P}( X(A_i(x))=0))\\ &\qquad = 1-\prod_{i=1}^{2d} (1- {\textrm{e}}^{-\gamma\varphi(U_i)x})\\ &\qquad \le 1-(1- {\textrm{e}}^{- \gamma ax})^{2d}\\ &\qquad \le 2d \,{\textrm{e}}^{-\gamma ax}, \end{align*}

by Bernoulli’s inequality. This was the assertion.

3. Proof of the upper bound

The approach to proving the right-hand estimate of (6) consists in establishing an extremal property of balls and then finding a connection to a known result on approximation of balls by convex hulls of finitely many random points. Since we are dealing with Poisson processes, this requires extra arguments in either step.

Let X be a stationary Poisson hyperplane process in ${\mathbb R}^d$ , with a non-degenerate spherical directional distribution $\varphi$ and with intensity $\gamma$ . If a convex body $K\in{{\mathcal K}}_d$ is given, we let $Z_K$ denote the K-cell defined by X and K, as in (4). In order to be able to compare $Z_K$ and $Z_L$ for different $K,L\in {{\mathcal K}}_d$ , we use an auxiliary Poisson process. For this, we consider the product space $E\colon={{\mathbb S}^{d-1}}\times[0,\infty)$ with the product measure $\varphi\otimes \lambda_+$ , where $\lambda_+$ is the Lebesgue measure on $[0,\infty)$ . Let Y be the Poisson process on E with intensity measure $2\gamma\varphi\otimes\lambda_+$ . (Its existence and uniqueness up to stochastic equivalence follows, for example, from [Reference Schneider and Weil27, Theorem 3.2.1].) Let M(E) denote the set of all locally finite subsets $S\subset E$ with the property that the set $\{u\in{{\mathbb S}^{d-1}} \colon (u,t)\in S \mbox{ for some }t\ge 0\}$ positively spans ${\mathbb R}^d$ . For $\eta\in{N}_s(E)$ with support in M(E), we define

$$P(\eta,K)\colon= \bigcap_{(u,t)\in{\textrm{supp}}\,\eta} H^-(u,h(K,u)+t)$$

for $K\in{{\mathcal K}}_d$ , where $h(K,\cdot)\colon=\max\{\langle x,\cdot\rangle \colon x\in K\}$ denotes the support function of K. This is a polytope containing K. We shall see below that the random polytope P(Y,K) is stochastically equivalent to the K-cell $Z_K$ defined by the hyperplane process X. We use the random polytopes P(Y,K) to show that the function $K\mapsto {{\mathbb E}} W(Z_K)$ is concave and continuous on ${{\mathcal K}}_d$ . Here a function $f \colon {{\mathcal K}}_d\to{\mathbb R}$ is called concave if

$$f((1-\alpha)K+\alpha L)\ge (1-\alpha)\,f(K)+\alpha f(L)$$

for $K,L\in{{\mathcal K}}_d$ and $\alpha\in[0,1]$ , where $K+L\colon=\{x+y \colon x\in K,\,y\in L\}$ and $\beta K\colon=\{\beta x \colon x\in K\}$ for $K,L\in {{\mathcal K}}_d$ and $\beta\ge 0$ .

Lemma 2. For $K,L\in{{\mathcal K}}_d$ and $\alpha\in[0,1]$ ,

(7) \begin{equation} {{\mathbb E}} W(Z_{(1-\alpha)K+\alpha L}) \ge (1-\alpha){{\mathbb E}} W(Z_K)+\alpha {{\mathbb E}} W(Z_L). \label{eqn7} \end{equation}

The functional $K\mapsto {{\mathbb E}} W(Z_K)$ is continuous on ${{\mathcal K}}_d$ .

Proof. For $\eta\in{N}_s(E)$ we have

$$(1-\alpha)P(\eta,K) +\alpha P(\eta,L) \subseteq P(\eta, (1-\alpha)K+\alpha L),$$

as follows immediately from the definition of $P(\eta,K)$ and the linearity properties of the support function. The monotonicity and linearity properties of the mean width yield

$$W(P(\eta, (1-\alpha)K+\alpha L)) \ge (1-\alpha) W(P(\eta,K))+\alpha W(P(\eta,L)).$$

Here we can replace $\eta$ with Y. Then linearity and monotonicity of the expectation yield

(8) \begin{equation} {{\mathbb E}} W(P(Y,(1-\alpha)K+\alpha L))\ge (1-\alpha){{\mathbb E}} W(P(Y,K)) +\alpha{{\mathbb E}} W(P(Y,L)). \label{eqn8} \end{equation}

We define the Poisson hyperplane process $X_K$ by

$$X_K(A)\colon= X(A\setminus {\mathcal{H}}_{{\textrm{int}}\,K})$$

for Borel sets $A\subset{\mathcal{H}}$ . Its intensity measure is given by

\begin{align*} {{\mathbb E}\,} X_K(A) &= \Theta(A\setminus {\mathcal{H}}_{{\textrm{int}}\,K})\\ &= \gamma\int_{{{\mathbb S}^{d-1}}} \int_{-\infty}^\infty {\bf 1}\{H(u,\tau)\in A\}{\bf 1}\{H(u,\tau)\cap{\textrm{int}}\,K=\emptyset\}\,{\textrm{d}}\tau\,\varphi({\textrm{d}} u)\\ &= \gamma\int_{{{\mathbb S}^{d-1}}} \int_{-\infty}^{-h(K,-u)} {\bf 1}\{H(u,\tau)\in A\}\,{\textrm{d}}\tau\,\varphi({\textrm{d}} u)\\ &\quad\, + \gamma\int_{{{\mathbb S}^{d-1}}} \int_{h(K,u)}^{\infty} {\bf 1}\{H(u,\tau)\in A\}\,{\textrm{d}}\tau\,\varphi({\textrm{d}} u)\\ &= 2\gamma\int_{{{\mathbb S}^{d-1}}} \int_{h(K,u)}^{\infty} {\bf 1}\{H(u,\tau)\in A\}\,{\textrm{d}}\tau\,\varphi({\textrm{d}} u), \end{align*}

where we have used the fact that $\varphi$ is an even measure. Next, we define a mapping $F_K \colon E\to {\mathcal{H}}$ by

$$F_K(u,t)\colon= H(u,h(K,u)+t)$$

and denote for $\eta\in{N}_s(E)$ by $F_K(\eta)$ the pushforward of $\eta$ under $F_K$ . Then $F_K(Y)$ is a Poisson hyperplane process. For its intensity measure we obtain, for Borel sets $A\subset{\mathcal{H}}$ ,

\begin{align*} {{\mathbb E}\,}(F_K(Y))(A) &= {{\mathbb E}} Y(F_K^{-1}(A))\\ &= 2\gamma \int_{{{\mathbb S}^{d-1}}} \int_0^\infty {\bf 1}\{(u,t)\in F_K^{-1}(A)\}\,{\textrm{d}} t\,\varphi({\textrm{d}} u)\\ &= 2\gamma \int_{{{\mathbb S}^{d-1}}} \int_0^\infty {\bf 1}\{H(u,h(K,u)+t)\in A\}\,{\textrm{d}} t\,\varphi({\textrm{d}} u)\\ &= 2\gamma \int_{{{\mathbb S}^{d-1}}} \int_{h(K,u)}^\infty {\bf 1}\{H(u,\tau)\in A\}\,{\textrm{d}} \tau\,\varphi({\textrm{d}} u). \end{align*}

Thus, $X_K$ and $F_K(Y)$ have the same intensity measure. Since either of them is a Poisson process, they are stochastically equivalent. It follows that the zero cell $Z_K$ is stochastically equivalent to the random polytope P(Y,K). Therefore, (8) yields the assertion (7).

To prove the continuity assertion, let $K,K_i\in{{\mathcal K}}_d$ for $i\in{\mathbb N}$ and suppose that $K_i\to K$ in the Hausdorff metric, as $i\to\infty$ . If $\varepsilon_0>0$ is small enough, then

$$K_\varepsilon\colon= \bigcap_{u\in{{\mathbb S}^{d-1}}} H^-(u,h(K,u)+\varepsilon)$$

is a convex body for any $\varepsilon >-\varepsilon_0$ . Clearly, $K_\varepsilon\to K$ as $\varepsilon\to 0$ . For given $\varepsilon>0$ , let i be so large that $K_{-\varepsilon}\subset K_i\subset K_\varepsilon$ . Observe also that $X({\mathcal{H}}_{K_i}\triangle{\mathcal{H}}_K)=0$ (where $\triangle$ denotes the symmetric difference) implies that $Z_{K_i}=Z_K$ . Therefore,

\begin{align*} |{{\mathbb E}} W(Z_{K_i})-{{\mathbb E}} W(Z_K)| &\le {{\mathbb E}}|W(Z_{K_i}) - W(Z_K)|\\ &\le {\mathbb P}(X({\mathcal{H}}_{K_i}\triangle {\mathcal{H}}_K)>0){{\mathbb E}} W(Z_{K_{\varepsilon}})\\ &\le {\mathbb P}(X({\mathcal{H}}_{K_\varepsilon}\setminus{\mathcal{H}}_{K_{-\varepsilon}})>0){{\mathbb E}} W(Z_{K_{\varepsilon_0}})\\ & \to 0 \end{align*}

as $\varepsilon\to 0$ . The limit follows from the fact that

$$\Theta({\mathcal{H}}_{K_\varepsilon}\setminus {\mathcal{H}}_{K_{-\varepsilon}}) =\gamma\int_{{{\mathbb S}^{d-1}}} [h(K_\varepsilon,u) -h(K_{-\varepsilon},u)]\varphi({\textrm{d}} u)\to 0$$

as $\varepsilon\to 0$ , by monotone convergence. This proves the continuity assertion.

From now on, we assume that the stationary Poisson hyperplane process X is isotropic and has intensity n. Then its intensity measure is given by $\Theta=n\mu$ with

(9) \begin{equation} \mu=\int_{{{\mathbb S}^{d-1}}} \int_{-\infty}^\infty {\bf 1}\{H(u,\tau)\in\cdot\}\,{\textrm{d}}\tau\,\sigma({\textrm{d}} u), \label{eqn9} \end{equation}

where $\sigma$ is the normalized spherical Lebesgue measure. For convex bodies $K,L\in{{\mathcal K}}_d$ with $K\subset L$ , we have

(10) \begin{align} \mu({\mathcal{H}}_L\setminus{\mathcal{H}}_K) &= \int_{{\mathcal{H}}\setminus{\mathcal{H}}_K} {\bf 1}\{H\cap L\not=\emptyset\}\,\mu({\textrm{d}} H)\nonumber\\ &= \int_{{{\mathbb S}^{d-1}}} \int_{-\infty}^\infty {\bf 1}\{H(u,\tau)\cap L\not=\emptyset\}{\bf 1}\{H(u,\tau)\cap K=\emptyset\}\,{\textrm{d}}\tau\,\sigma({\textrm{d}} u)\nonumber\\ &= 2\int_{{{\mathbb S}^{d-1}}}[h(L,u)-h(K,u)]\,\sigma({\textrm{d}} u)\nonumber\\ &= W(L)-W(K). \label{eqn10}\end{align}

Lemma 3. If X is isotropic, then the functional

$$K\mapsto \dfrac{{{\mathbb E}} W(Z_K)}{W(K)},\quad K\in{{\mathcal K}}_d,$$

attains its maximum at balls.

Proof. If X is isotropic, then the functional $K\mapsto {{\mathbb E}} W(Z_K)$ is invariant under rigid motions. Since by Lemma 2 it is concave and continuous on ${{\mathcal K}}_d$ , it is well known that on the set of convex bodies $K\in {{\mathcal K}}_d$ with given mean width W(K) it attains its maximum at balls. The proof, which uses Hadwiger’s ‘Zweites Kugelungstheorem’ ([Reference Hadwiger12, pp. 170–171], reproduced in [Reference Schneider25, Theorem 3.3.5]), is carried out in [Reference Böröczky and Schneider6, p. 621].

To take advantage of the preceding lemma, we connect this to a known asymptotic result about convex hulls of i.i.d. random points in a ball. First we write the result of Lemma 3 in the form

(11) \begin{equation} {{\mathbb E}} W(Z_K) - W(K) \ll {{\mathbb E}} W(Z_{B^d})-W(B^d). \label{eqn11} \end{equation}

We recall that here $Z_K=Z_K^{(n)}$ and $Z_{B^d}=Z_{B^d}^{(n)}$ and that we intend to let n tend to infinity. In view of this, we choose a number $R>b$ , where b is the constant appearing in Lemma 1 for $K=B^d$ , and state that

(12) \begin{equation} {{\mathbb E}} W(Z_{B^d}) - {{\mathbb E}\,} [W(Z_{B^d}){\bf 1}\{R_o(Z_{B^d})<R\}]={\textrm{O}}(n^{-1}) \label{eqn12} \end{equation}

as $n\to \infty$ (where the constant involved in ${\textrm{O}}$ depends on R). For the proof, we note that the left side of (12) can be estimated by

\begin{align*} {{\mathbb E}\,} [W(Z_{B^d}){\bf 1}\{R_o(Z_{B^d})\ge R\}] & \le {{\mathbb E}\,} [2R_o(Z_{B^d}){\bf 1}\{R_o(Z_{B^d})\ge R\}]\\ &= 2\int_\Omega R_o(Z_{B^d}){\bf 1}\{R_o(Z_{B^d})\ge R\}\,{\textrm{d}}{\mathbb P}\\ &= 2\int_0^\infty {\mathbb P}(R_o(Z_{B^d}){\bf 1}\{R_o(Z_{B^d})\ge R\}>t)\,{\textrm{d}} t\\ &= 2R\,{\mathbb P}(R_o(Z_{B^d})\ge R) + 2\int_R^\infty {\mathbb P}(R_o(Z_{B^d})>t)\,{\textrm{d}} t. \end{align*}

Lemma 1 provides an estimate for ${\mathbb P}(R_o(Z_{B^d})\ge b(1+x))$ . Using this with $b(1+x)=R$ for the first summand and with $b(1+x)=t$ for the second summand (and observing that now $\gamma=n$ ), we obtain (12).

We use the bijective mapping

(13) \begin{equation} \xi \colon {\mathcal{H}}\setminus {\mathcal{H}}_{\{o\}} \to {\mathbb R}^d\setminus\{o\},\quad \xi(H(u,\tau))= \tau^{-1}u. \label{eqn13} \end{equation}

Let $\kappa_0$ be the pushforward of the measure $\mu$ , restricted to ${\mathcal{H}}\setminus {\mathcal{H}}_{B^d}$ , under $\xi$ . Then

$$\label{3.5a} \kappa_0(A) =\dfrac{2}{\omega_d}\int_A \|x\|^{-(d+1)}\lambda_d({\textrm{d}} x) $$

for Borel sets $A\subset B^d\setminus\{o\}$ , where $\omega_d$ is the surface area of the unit sphere. The measure $\kappa_0$ is infinite, but finite on compact subsets of $B^d\setminus\{o\}$ .

Let $Y_n$ denote the Poisson point process in ${\mathbb R}^d$ with intensity measure $n\kappa_0$ . Let $Q_n$ be the convex hull of $Y_n$ . Then $Q_n$ is a random polytope, which is stochastically equivalent to the polar of $Z_{B^d}$ . With the constant $R>b$ chosen above, we set $r=1/R$ and $B_r= rB^d$ . By (10), we have

$$W(Z_{B^d})-W(B^d) = \int_{{\mathcal{H}}\setminus {\mathcal{H}}_{B^d}} {\bf 1}\{H\cap Z_{B^d}\not=\emptyset\}\,\mu({\textrm{d}} H)=\kappa_0(B^d\setminus Q_n),$$

hence

$${{\mathbb E}\,}[(W(Z_{B^d})-W(B^d)){\bf 1}\{R_o(Z_{B^d})<R\}]= {{\mathbb E}\,}[\kappa_0(B^d\setminus Q_n){\bf 1}\{B_r\subset Q_n\}]. $$

Now it follows from (11) and (12) that

(14) \begin{equation} {{\mathbb E}} W(Z_K)-W(K) \ll {{\mathbb E}\,}[\kappa_0(B^d\setminus Q_n){\bf 1}\{B_r\subset Q_n\}]+{\textrm{O}}(n^{-1}). \label{eqn14} \end{equation}

To express the latter expectation in a suitable way, we note that $Q_n$ is almost surely a simplicial polytope, hence each of its facets is the convex hull of d points of $Y_n$ . For any d points $x_1,\dots,x_d\in Y_n$ (almost surely, they are affinely independent and their affine hull does not contain o), we define

$$S(x_1,\dots,x_d)\colon= B^d\setminus H^-(x_1,\dots,x_d),$$

where $H^-(x_1,\dots,x_d)$ is the closed half-space bounded by ${\textrm{aff}}\{x_1,\dots,x_d\}$ that contains o. Further, we define

$$T(x_1,\dots,x_d)\colon= S(x_1,\dots,x_d) \cap {\textrm{pos}}\{x_1,\dots,x_d\}.$$

Then we have

\begin{align*} & \kappa_0(B^d\setminus Q_n){\bf 1}\{B_r\subset Q_n\} \\ & \qquad = \dfrac{1}{d!}\sum_{(x_1,\dots,x_d)\in (Y_n)^d_{\not=}} {\bf 1}\{Y_n(S(x_1,\dots,x_d))=0\}\kappa_0(T(x_1,\dots,x_d)){\bf 1}\{B_r\subset Q_n\}, \end{align*}

where $\eta^d_{\not=}$ denotes the set of ordered d-tuples of pairwise different elements from the support of $\eta$ . We note that if $B_r\subset Q_n$ , then points $x_1,\dots,x_d\in Y_n$ with $Y_n(S(x_1,\dots,x_d))=0$ automatically satisfy $x_1,\dots,x_d\in B^d\setminus B_r$ and ${\textrm{aff}}\{x_1,\dots,x_d\}\cap B_r=\emptyset$ a.s. Therefore,

\begin{align*} & \kappa_0(B^d\setminus Q_n){\bf 1}\{B_r\subset Q_n\}\\ &\qquad = \dfrac{1}{d!}\sum_{(x_1,\dots,x_d)\in (Y_n)^d_{\not=}} {\bf 1}\{Y_n(S(x_1,\dots,x_d))=0,\,B_r\subset Q_n\}\kappa_0(T(x_1,\dots,x_d))\\ &\qquad\quad\, \times\,{\bf 1}\{x_1,\dots,x_d\in B^d\setminus B_r\}{\bf 1}\{{\textrm{aff}}\{x_1,\dots,x_d\}\cap B_r=\emptyset\}. \end{align*}

Using the Slivnyak–Mecke formula (see e.g. [Reference Schneider and Weil27, Corollary 3.2.3]) and noting that $n\kappa_0$ is the intensity measure of $Y_n$ , we obtain

\begin{align*} & {{\mathbb E}\,}[\kappa_0(B^d\setminus Q_n){\bf 1}\{B_r\subset Q_n\}]\\ &\qquad = \dfrac{n^d}{d!}\int_{B^d\setminus B_r}\dots\int_{B^d\setminus B_r} {{\mathbb E}} {\bf 1}\{Y_n(S(x_1,\dots,x_d))=0,\,B_r\subset {\textrm{conv}}(Y_n\cup\{x_1,\dots,x_d\})\}\\ &\qquad\quad\, \times \kappa_0(T(x_1,\dots,x_d)){\bf 1}\{{\textrm{aff}}\{x_1,\dots,x_d\}\cap B_r=\emptyset\}\,\kappa_0({\textrm{d}} x_1)\cdots\kappa_0({\textrm{d}} x_d). \end{align*}

Let $\lambda_0\colon= (2/\omega_d)\lambda_d$ . For Borel sets $A\subset B^d\setminus B_r$ we have

$$\lambda_0(A)\le \kappa_0(A)\le r^{-(d+1)}\lambda_0(A).$$

For fixed $x_1,\dots,x_d\in B^d\setminus B_r$ with ${\textrm{aff}}\{x_1,\dots,x_d\}\cap B_r=\emptyset$ , we have

\begin{align*} & {{\mathbb E}} {\bf 1}\{Y_n(S(x_1,\dots,x_d))=0,\,B_r\subset {\textrm{conv}}(Y_n\cup\{x_1,\dots,x_d\}\}\\ &\qquad \le {{\mathbb E}} {\bf 1}\{Y_n(S(x_1,\dots,x_d))=0\}\\ &\qquad = {\textrm{e}}^{-n\kappa_0(S(x_1,\dots,x_d))}\\ &\qquad \le {\textrm{e}}^{-n\lambda_0(S(x_1,\dots,x_d))}. \end{align*}

Therefore, we can estimate

\begin{align*} & {{\mathbb E}\,}[\kappa_0(B^d\setminus Q_n){\bf 1}\{B_r\subset Q_n\}] \\ &\qquad \ll n^d\int_{B^d}\dots\int_{B^d} \,{\textrm{e}}^{-n\lambda_0(S(x_1,\dots,x_d))} \,\lambda_0(T(x_1,\dots,x_d))\,\lambda_0({\textrm{d}} x_1)\cdots\lambda_0({\textrm{d}} x_d). \end{align*}

Let $\widetilde Y_n$ be a Poisson point process in ${\mathbb R}^d$ with intensity measure $n\lambda_0$ , and let

$$\Pi_n\colon= {\textrm{conv}}(\widetilde Y_n\cap B^d).$$

Using the Slivnyak–Mecke formula as above yields that

\begin{align*} & {{\mathbb E}}\lambda_0(B^d\setminus \Pi_n)\\ &\qquad = n^d \int_{B^d}\dots\int_{B^d} \,{\textrm{e}}^{-n\lambda_0(S(x_1,\dots,x_d))} \lambda_0(T(x_1,\dots,x_d))\,\lambda_0({\textrm{d}} x_1)\cdots\lambda_0({\textrm{d}} x_d). \end{align*}

We conclude that

$${{\mathbb E}\,}[\kappa_0(B^d\setminus Q_n){\bf 1}\{B_r\subset Q_n\}] \ll {{\mathbb E}} \lambda_0(B^d\setminus \Pi_n).$$

It follows from Lemma 1 in [Reference Reitzner18] that

$${{\mathbb E}} \lambda_0(B^d\setminus\Pi_n) \ll n^{-2/(d+1)}.$$

Together with (14), this yields the upper bound in (6).

4. Proof of the lower bound

The proof of the left-hand estimate of (6) requires only a few changes in the proof of the corresponding inequality in [Reference Böröczky and Schneider6, (1.3)].

Let $K\in{{\mathcal K}}_d$ . For $x\in{\mathbb R}^d\setminus K$ , we define $K^x\colon= {\textrm{conv}}(K\cup \{x\})$ and set

$$m(H)\colon= \min\{W(K^x)-W(K){\colon}x\in H\}$$

for hyperplanes $H\in{\mathcal{H}}\setminus {\mathcal{H}}_K$ . For $t> 0$ we define

$${\mathcal{H}}_K(t) \colon= \{H\in {\mathcal{H}}\setminus {\mathcal{H}}_K{\colon} m(H)\le t\}.$$

We assume now, as in Theorem 1, that X is a stationary and isotropic Poisson hyperplane process in ${\mathbb R}^d$ of intensity $n\in {\mathbb N}$ . Let $H\in{\mathcal{H}}\setminus{\mathcal{H}}_K$ , and let $z\in H$ be such that $m(H)= W(K^{z})-W(K)$ (clearly, such a point exists). If no hyperplane of X separates z and K, then $z\in Z_K^{(n)}$ and hence $H\cap Z_K^{(n)}\not=\emptyset$ . It follows that

\begin{align*} {\mathbb P} (H\cap Z_K^{(n)}\not=\emptyset ) &\ge {\mathbb P}(X({\mathcal{H}}_{K^{z}}\setminus{\mathcal{H}}_K)=0)\\ &={\rm exp}\, [-\Theta({\mathcal{H}}_{K^{z}}\setminus{\mathcal{H}}_K)]\\ & = {\textrm{e}}^{-nm(H)}, \end{align*}

where (10) was used. Therefore, we obtain (using (10) again and Fubini’s theorem)

\begin{align*} {{\mathbb E}} W(Z_K^{(n)})-W(K) &{} = \int_\Omega\int_{{\mathcal{H}}\setminus{\mathcal{H}}_K} {\bf 1}\{H\cap Z_K^{(n)}\not=\emptyset\}\,\mu({\textrm{d}} H)\,{\textrm{d}}{\mathbb P}\\ &= \int_{{\mathcal{H}}\setminus{\mathcal{H}}_K} {\mathbb P}(H\cap Z_K^{(n)}\not=\emptyset)\,\mu({\textrm{d}} H)\\ &\ge \int_{{\mathcal{H}}\setminus{\mathcal{H}}_K} \,{\textrm{e}}^{-nm(H)}\,\mu({\textrm{d}} H)\\ &\ge \int_{{\mathcal{H}}\setminus {\mathcal{H}}_K} {\bf 1}\{m(H)\le t\} \,{\textrm{e}}^{-nt}\,\mu({\textrm{d}} H)\\ &= {\textrm{e}}^{-nt}\mu({\mathcal{H}}_K(t)), \end{align*}

where $t> 0$ can be any number. The choice $t=1/n$ gives

(15) \begin{equation} {{\mathbb E}} W(Z_K^{(n)})-W(K) \ge {\textrm{e}}^{-1}\mu({\mathcal{H}}_K(1/n)). \label{eqn15} \end{equation}

The remainder of the proof is similar to that in [6, p. 619]. For the reader’s convenience, we adapt the argument to the present case. The idea is to use a result from Bárány and Larman [Reference Bárány and Larman4], together with polarity.

We translate K so that (say) its centroid is at the origin. Let $K^\circ$ be the polar body of K. For $x\in K^\circ$ , let v(x) denote the minimal volume that a closed half-space containing x cuts off from $K^\circ$ , and define

$$\label{4.2} K^\circ(t)\colon= \{x\in K^\circ \colon v(x)\le t\} $$

for $t> 0$ . We can choose $\rho,t_0>0$ such that $\rho B^d\subset K^\circ(t)$ for $0<t\le t_0$ .

Let $\psi{\colon} {\mathbb R}^d\setminus\{o\}\to {\mathcal{H}}\setminus{\mathcal{H}}_{\{o\}}$ be the inverse of the map $\xi$ defined by (13) ; thus

$$\psi(ru) = H(u,r^{-1})\quad\mbox{for } u\in{{\mathbb S}^{d-1}},\,r> 0.$$

Let $\nu$ denote the pushforward of the Lebesgue measure $\lambda_d$ under $\psi$ ; thus

(16) \begin{equation} \nu(A) = \omega_d\int_{{{\mathbb S}^{d-1}}}\int_0^\infty {\bf 1}\{H(u,\tau)\in A\}\tau^{-(d+1)}{\textrm{d}}\tau\,\sigma({\textrm{d}} u) \label{eqn16} \end{equation}

for Borel sets $A\subset {\mathcal{H}}\setminus{\mathcal{H}}_{\{o\}}$ .

Let $H(u,\tau)$ be a hyperplane contained in ${\mathcal{H}}_{\rho^{-1}B^d}\setminus {\mathcal{H}}_K$ . Since $H(u,\tau)\cap K=\emptyset$ , we have $\tau\ge h(K,u)$ , which is bounded from below by a positive constant depending only on K. Since $H(u,\tau)\cap \rho^{-1}B^d\not=\emptyset$ , we have $\tau\le \rho^{-1}$ . Now comparison of (16) and (9) yields the existence of constants $c_1,c_2>0$ , depending only on d and K, such that

$$c_1\nu(A)\le \mu(A) \le c_2\nu(A) \quad \mbox{if } A\subset {\mathcal{H}}_{\rho^{-1}B^d}\setminus {\mathcal{H}}_K.$$

Let $0<t\le t_0$ and $x\in K^\circ(t)\setminus\partial K^\circ$ . There is a hyperplane E through x that bounds a closed half-space $E^+$ not containing o, such that $\lambda_d(K^\circ\cap E^+)\le t$ . If $H\colon=\psi(x)$ and $y \colon=\psi^{-1}(E)$ , then $y\in H\in {\mathcal{H}}_{\rho^{-1}B^d}\setminus {\mathcal{H}}_K$ . The mapping $\psi$ maps the cap $K^\circ\cap E^+$ bijectively onto the set of hyperplanes (weakly) separating y and K. We denote this set by ${\mathcal{H}}^y_K$ . It follows that

$$m(H) \le W(K^y)-W(K)=\mu({\mathcal{H}}_K^y) \le c_2\nu({\mathcal{H}}_K^y) = c_2\lambda_d(K^\circ\cap E^+) \le c_2t,$$

hence $H\in {\mathcal{H}}_K(c_2t)$ . Since $x\in K^\circ(t)\setminus\partial K^\circ$ was arbitrary, this shows that $\psi(K^\circ(t)) \subset {\mathcal{H}}_K(c_2t)$ . Therefore,

$$\lambda_d(K^\circ(t)) =\nu(\psi(K^\circ(t))\le \nu({\mathcal{H}}_K(c_2t)) \le c_1^{-1}\mu({\mathcal{H}}_k(c_2t)).$$

If we choose $t=1/(c_2n)$ , then this inequality together with (15) shows that

$${{\mathbb E}} W(Z_K^{(n)}) - W(K) \ge {\textrm{e}}^{-1}\mu({\mathcal{H}}_K(1/n))\ge {\textrm{e}}^{-1}c_1\lambda_d(K^\circ((c_2n)^{-1}).$$

Theorem 2 of [Reference Bárány and Larman4] says that

$$\lambda_d(K^\circ(\varepsilon)) \ge c\varepsilon\log^{d-1}(1/\varepsilon)$$

for sufficiently small $\varepsilon>0$ , with some constant c depending only on K. It follows that

$$ {{\mathbb E}} W(Z_K^{(n)}) - W(K) \ge c_3 n^{-1}\log^{d-1} n$$

for all sufficiently large n, where $c_3$ is a constant depending only on d and K. By adapting the constant, we can assume that this holds for all $n\in {\mathbb N}$ . This completes the proof of the lower bound in (6).