Rigidity of Riemannian embeddings of discrete metric spaces

Abstract

Let M be a complete, connected Riemannian surface and suppose that ${\mathcal {S}}\subset M$ is a discrete subset. What can we learn about M from the knowledge of all Riemannian distances between pairs of points of ${\mathcal {S}}$? We prove that if the distances in ${\mathcal {S}}$ correspond to the distances in a 2-dimensional lattice, or more generally in an arbitrary net in ${\mathbb {R}}^2$, then M is isometric to the Euclidean plane. We thus find that Riemannian embeddings of certain discrete metric spaces are rather rigid. A corollary is that a subset of ${\mathbb {Z}}^3$ that strictly contains ${\mathbb {Z}}^2 \times \{ 0 \}$ cannot be isometrically embedded in any complete Riemannian surface.

Appendix: Continuity of area

Suppose that for any $m \ge 1$ we are given a metric $d_m$ on ${\mathbb {R}}^2$, such that the following hold:

• (i) For any m, the metric $d_m$ induces the standard topology on ${\mathbb {R}}^2$.

• (ii) For any $x,y \in {\mathbb {R}}^2$ we have $d_m(x,y) \longrightarrow |x-y|$ as $m \rightarrow \infty$, and the convergence is locally uniform.

• (iii) For any m, the metric space $({\mathbb {R}}^2, d_m)$ is isometric to a complete, connected, 2-dimensional Riemannian manifold in which all geodesics are minimizing.

Write $\text {Area}_m$ for the Riemannian area measure on ${\mathbb {R}}^2$ that corresponds to $d_m$ under the above isometry.

Proposition 7.1

For $D = D(0,1) =\{ x \in {\mathbb {R}}^2 \, ; \, |x| < 1 \}$ we have $\text {Area}_m(D) \longrightarrow \pi$ as $m \rightarrow \infty$.

We were unable to find a proof of Proposition 7.1 in the literature, even though Ivanov’s paper [22] contains a deeper result which “almost” implies this proposition. A proof of Proposition 7.1 is thus provided in this Appendix.

By a $d_m$-geodesic in ${\mathbb {R}}^2$ we mean a geodesic with respect to the metric $d_m$. The $d_m$-length of a $d_m$-rectifiable curve $\gamma$ is denoted by $\text {Length}_m(\gamma )$. A set $K \subseteq {\mathbb {R}}^2$ is $d_m$-convex if the intersection of any $d_m$-geodesic with K is connected. All $d_m$-geodesics are minimizing, and each complete $d_m$-geodesic divides ${\mathbb {R}}^2$ into two connected components. Each of these connected components is a $d_m$-convex, open set called a $d_m$-half-plane. The intersection of finitely many $d_m$-half-planes, if bounded and non-empty, is called a $d_m$-polygon. Note that our polygons are always open and convex. The boundary of any $d_m$-polygon consists of finitely many vertices and the same number of edges, and each edge is a $d_m$-geodesic segment.

Write ${\mathcal {G}}_m$ for the collection of all complete $d_m$-geodesics in ${\mathbb {R}}^2$, where we identify between two geodesics if they differ by an orientation-preserving reparametrization. Write $\sigma _m$ for the Liouville (or étendue) measure on ${\mathcal {G}}_m$, see Kloeckner and Kuperberg [24, Section 5.2] and Álvarez-Paiva and Berck [1, Section 5]) and references therein for the basic properties of this measure, and for the formulae of Santaló and Crofton from integral geometry. The Santaló formula implies that for any open set $U \subseteq {\mathbb {R}}^2$,

$$\begin{aligned} \text {Area}_m( U ) = \frac{1}{2 \pi } \int _{{\mathcal {G}}_m} \text {Length}_m(\gamma \cap U ) d \sigma _m(\gamma ). \end{aligned}$$

(We remark that $\gamma \cap U$ can be disconnected, yet it is a disjoint, countable union of $d_m$-geodesics, and $\text {Length}_m(\gamma \cap U)$ is the sum of the $d_m$-lengths of these $d_m$-geodesics). The Crofton formula implies that for any $d_m$-polygon $P \subseteq {\mathbb {R}}^2$,

$$\begin{aligned} \text {Perimeter}_m( P ) = \frac{1}{2} \cdot \sigma _m \left( \left\{ \gamma \in {\mathcal {G}}_m \, ; \, \gamma \cap P \ne \emptyset \right\} \right) , \end{aligned}$$

where $\text {Perimeter}_m(P) = \text {Length}_m(\partial P)$. When we discuss $\text {Area}, \text {Length}, \text {Perimeter}$ or polygons without the subscript m we refer to the usual Euclidean geometry in ${\mathbb {R}}^2$. Write ${\mathcal {G}}$ for the collection of all lines in ${\mathbb {R}}^2$, where we identify between two lines if they differ by an orientation-preserving reparametrization. Write $\sigma$ for the Euclidean Liouville measure on ${\mathcal {G}}$. We require the following Euclidean lemma:

Lemma 7.2

Let ${\tilde{\sigma }}$ be a Borel measure on ${\mathcal {G}}$ such that for any convex polygon $P \subseteq D(0,2) \subseteq {\mathbb {R}}^2$,

$$\begin{aligned} \mathrm{perimeter}( P ) = \frac{1}{2} \cdot {\tilde{\sigma }} \left( \left\{ \ell \in {\mathcal {G}}\, ; \, \ell \cap P \ne \emptyset \right\} \right) . \end{aligned}$$

(74)

Then

$$\begin{aligned} \frac{1}{2 \pi } \int _{{\mathcal {G}}} \mathrm{length}(\ell \cap D ) d {\tilde{\sigma }}(\ell ) = \mathrm{area}(D) = \pi . \end{aligned}$$

(75)

Proof

For any rotation $U \in SO(2)$, the perimeter of the rotated polygon $U(P) \subset {\mathbb {R}}^2$ is the same as the perimeter of P. Hence formula (74) holds true with ${\tilde{\sigma }}$ replaced by $U_* {\tilde{\sigma }}$, where by $U_* {\tilde{\sigma }}$ we mean the push-forward of ${\tilde{\sigma }}$ under the map U acting on ${\mathcal {G}}$ by rotating lines. Moreover, since $U(D) = D$, replacing ${\tilde{\sigma }}$ by $U_* {\tilde{\sigma }}$ does not change the value of the integral on the left-hand side of (75).

We may thus replace the measure ${\tilde{\sigma }}$ by the average of $U_* {\tilde{\sigma }}$ over $U \in SO(2)$, and assume from now on that ${\tilde{\sigma }}$ is a rotationally-invariant measure on ${\mathcal {G}}$. The validity of (74) for any convex polygon $P \subset D(0,2)$ implies its validity for all convex sets in the disc D(0, 2). Indeed, both the left-hand side and the right-hand side of (74) are monotone in the convex set P under inclusion, and convex polygons are dense in the class of all convex subsets of D(0, 2). Consequently, for any $0< \rho < 2$,

$$\begin{aligned} 2 \pi \rho = \text {Perimeter}(D(0,\rho )) = \frac{1}{2} \cdot {\tilde{\sigma }} \left( \left\{ \ell \in {\mathcal {G}}\, ; \, \ell \cap D(0,\rho ) \ne \emptyset \right\} \right) . \end{aligned}$$

(76)

For $\ell \in {\mathcal {G}}$ write $r(\ell ) = \inf _{x \in \ell } |x| \in [0, \infty )$. We may reformulate (76) as follows:

$$\begin{aligned} {\tilde{\sigma }} \left( \left\{ \ell \in {\mathcal {G}}\, ; \, r(\ell )< \rho \right\} \right) = \sigma \left( \left\{ \ell \in {\mathcal {G}}\, ; \, r(\ell )< \rho \right\} \right) \qquad \text{ for } \text{ any } 0< \rho < 2. \qquad \end{aligned}$$

(77)

Since both $\sigma$ and ${\tilde{\sigma }}$ are rotationally-invariant measures on ${\mathcal {G}}$, they are completely determined by their push-forward under the map $\ell \mapsto r(\ell )$. From (77) we learn that ${\tilde{\sigma }}$ coincides with $\sigma$ on the set ${\mathcal {G}}\cap r^{-1}([0,2))$. By the Satanló formula for $\sigma$,

$$\begin{aligned} \frac{1}{2\pi } \int _{{\mathcal {G}}} \text {Length}(\ell \cap D) d {\tilde{\sigma }}(\ell ) =\frac{1}{2\pi } \int _{{\mathcal {G}}} \text {Length}(\ell \cap D) d \sigma (\ell ) = \text {Area}(D) = \pi . \end{aligned}$$

$\square$

When we refer to the Hausdorff metric below, we always mean the Euclidean Hausdorff metric (the Hausdorff metric is defined, e.g., in [6]). Write $(x,y) \subseteq {\mathbb {R}}^2$ for the Euclidean interval between $x,y \in {\mathbb {R}}^2$ excluding the endpoints, and $[x,y] = (x,y) \cup \{ x,y \}$. We similarly write $[x,y]_m$ and $(x,y)_m$ for the $d_m$-geodesic between x and y, with and without the endpoints. We claim that for any $x,y \in {\mathbb {R}}^2$,

$$\begin{aligned}{}[x,y]_m \xrightarrow {m \rightarrow \infty } [x,y] \end{aligned}$$

(78)

in the Hausdorff metric. Indeed, for any $0 \le \lambda \le 1$, the point on $[x,y]_m$ whose $d_m$-distance from x equals $\lambda \cdot d_m(x,y)$ must converge to the point on [x, y] whose Euclidean distance from x equals $\lambda \cdot |x-y|$. It follows from our assumptions that the convergence is uniform in $\lambda$, and (78) follows. Moreover, it follows that the Hausdorff convergence in (78) is locally uniform in $x,y \in {\mathbb {R}}^2$.

Write ${\overline{A}}$ for the closure of a set A. The Euclidean $\varepsilon$-neighborhood of a subset $A \subseteq {\mathbb {R}}^2$ is the collection of all $x \in {\mathbb {R}}^2$ with $d(x,A) < \varepsilon$ where $d(x,A) = \inf _{y \in A} |x-y|$. Given a convex polygon $P \subseteq {\mathbb {R}}^2$, for a sufficiently large m we define $P^{(m)} \subseteq {\mathbb {R}}^2$ to be the $d_m$-polygon with the same vertices as P. We need m to be sufficiently large in order to guarantee that no vertex of P is in the $d_m$-convex hull of the other vertices.

Lemma 7.3

Let $P_0, P_1 \subseteq D(0,3)$ be convex polygons such that $\overline{P_0} \subseteq P_1$. Then there exist $m_0 \ge 1$ and $\varepsilon > 0$ such that the following holds: For any $m \ge m_0$ and any $x, x',y,y' \in D(0,3)$ with $|x -x'| < \varepsilon$ and $|y - y'| < \varepsilon$,

$$\begin{aligned} (x,y) \cap P_0 \ne \emptyset \qquad \Longrightarrow \qquad (x',y')_m \cap P_1^{(m)} \ne \emptyset , \end{aligned}$$

and

$$\begin{aligned} (x',y')_m \cap P_0^{(m)} \ne \emptyset \qquad \Longrightarrow \qquad (x,y) \cap P_1 \ne \emptyset . \end{aligned}$$

Proof

From the Hausdorff convergence in (78) it follows that for a sufficiently large m, the closure of $P_0^{(m)} \cup P_0$ is contained in $P_1 \cap P_1^{(m)}$. In fact, there exist $\delta > 0$ and $m_1 \ge 1$ such that for $m \ge m_1$, the Euclidean $\delta$-neighborhood of $P_0^{(m)} \cup P_0$ is contained in $P_1 \cap P_1^{(m)}$.

Set $\varepsilon = \delta / 2$. Since the convergence in (78) is uniform in $x,y \in D(0,3)$, there exists $m_0 \ge m_1$ such that for any $m \ge m_0$ and $x',y'\in D(0,3)$, the Hausdorff distance between $(x',y')$ and $(x',y')_{m}$ is at most $\varepsilon$. Thus for any $m \ge m_0$ and $x, x',y,y' \in D(0,3)$ with $|x -x'| < \varepsilon$ and $|y - y'| < \varepsilon$, the Hausdorff distance between (x, y) and $(x',y')$ is at most $\varepsilon$, and by the triangle inequality, the Hausdorff distance between (x, y) and $(x',y')_{m}$ is at most $2\varepsilon = \delta$.

Hence if (x, y) intersects $P_0$, then $(x',y')_m$ intersects the Euclidean $\delta$-neighborhood of $P_0$, which is contained in $P_1^{(m)}$. Similarly, if $(x',y')_m$ intersects $P_0^{(m)}$, then (x, y) intersects the $\delta$-neighborhood of $P_0^{(m)}$ which is contained in $P_1$. $\square$

Lemma 7.4

Let $K \subseteq {\mathbb {R}}^2$ be a bounded, open, convex set. Then there exist $d_m$-polygons $K_m^{\pm } \subseteq {\mathbb {R}}^2$ for $m \ge 1$, real numbers $\varepsilon _m \searrow 0$ and $m_0 \ge 1$, such that for any $m \ge m_0$ the following hold:

$$\begin{aligned} K_m^- \subseteq K \subseteq K_m^+, \end{aligned}$$

both boundaries $\partial K_m^{\pm }$ are $\varepsilon _m$-close to $\partial K$ in the Hausdorff metric, and the $d_m$-perimeters of $K_m^{\pm }$ differ from $\text {Perimeter}(K)$ by at most $\varepsilon _m$.

Proof

It suffices to show that for any fixed $\varepsilon > 0$ there exist $m_0 \ge 1$ and $d_m$-polygons $K_m^\pm \subset {\mathbb {R}}^2$, defined for any $m \ge m_0$, such that

$$\begin{aligned} K_m^- \subseteq K \subseteq K_m^+, \end{aligned}$$

both boundaries $\partial K_m^{\pm }$ are $\varepsilon$-close to $\partial K$ in the Hausdorff metric, and the $d_m$-perimeters of $K_m^{\pm }$ differ from $\text {Perimeter}(K)$ by at most $\varepsilon$.

Fix $\varepsilon > 0$. We may pick finitely many points in $\partial K$, cyclically ordered, such that when connecting each point via a segment to its two adjacent points, the result is a convex polygon whose boundary is $(\varepsilon /2)$-close to $\partial K$ in the Hausdorff metric. We may also require that the perimeter of this convex polygon differs from $\text {Perimeter}(K)$ by at most $\varepsilon /2$.

We slightly move these finitely many points inside K, and replace the segments between the points by $d_m$-geodesics. For a sufficiently large m, this defines a $d_m$-polygon $K_m^-$. It follows from (78) that for a sufficiently large m, the boundary $\partial K_m^-$ is $\varepsilon$-close to $\partial K$ in the Hausdorff metric, the $d_m$-perimeter of $K_m^-$ differs from $\text {Perimeter}(K)$ by at most $\varepsilon$, and $K_m^- \subseteq K$.

We still need to construct $K_m^+$. Approximate K by a convex polygon containing the closure of K in its interior, whose boundary is $(\varepsilon /2)$-close to $\partial K$ in the Hausdorff metric, and whose perimeter differs from $\text {Perimeter}(K)$ by at most $\varepsilon /2$. Replace the edges of this polygon by $d_m$-geodesics in order to form $K_m^+$. It follows from (78) that for a sufficiently large m, the $d_m$-convex set $K_m^+$ has the desired properties. $\square$

We apply Lemma 7.4 for the unit disc $D \subseteq {\mathbb {R}}^2$, and obtain two $d_m$-polygons $D_m^{\pm }$ with $D_m^- \subseteq D \subseteq D_m^+$ that satisfy the conclusions of the lemma. It follows from (78) that for $x,y \in {\mathbb {R}}^2$,

$$\begin{aligned} \text {Length}_m( (x,y)_m \cap D_m^{\pm } ) \xrightarrow {m \rightarrow \infty } \text {Length}( (x,y) \cap D ), \end{aligned}$$

(79)

and the convergence is locally uniform in $x, y \in {\mathbb {R}}^2$. Let us fix a convex polygon $K \subseteq {\mathbb {R}}^2$ such that $\overline{D(0,2)} \subseteq K$ and ${\overline{K}} \subseteq D(0,3)$. We apply Lemma 7.4 and obtain $d_m$-polygons $K_m = K_m^+$ for $m \ge 1$ that approximate K. For a set $A \subseteq {\mathbb {R}}^2$ denote

$$\begin{aligned} {\mathcal {G}}(A) = \{ \ell \in {\mathcal {G}}\, ; \, \ell \cap A \ne \emptyset \} \qquad \text {and} \qquad {\mathcal {G}}_{m}(A) = \{ \gamma \in {\mathcal {G}}_{m} \, ; \, \gamma \cap A \ne \emptyset \}. \end{aligned}$$

Definition 7.5

Define the map $T_K :{\mathcal {G}}(K) \rightarrow \partial K \times \partial K \subset {\mathbb {R}}^2 \times {\mathbb {R}}^2$ by

$$\begin{aligned} T_K(\ell ) = (a(\ell ), b(\ell )) \in \partial K \times \partial K, \end{aligned}$$

where $\ell \cap \partial K = \{ a(\ell ), b(\ell ) \}$ and the line $\ell$ is oriented from the point $a(\ell )$ towards the point $b(\ell )$. We analogously define the map $T_{m}: {\mathcal {G}}_m(K_m) \rightarrow \partial K_m \times \partial K_m \subset {\mathbb {R}}^2 \times {\mathbb {R}}^2$ via

$$\begin{aligned} T_m(\gamma ) = (a(\gamma ), b(\gamma )) \in \partial K_m \times \partial K_m, \end{aligned}$$

where $\gamma \cap \partial K_m = \{ a(\gamma ), b(\gamma ) \}$ and the geodesic $\gamma$ is oriented from $a(\gamma )$ towards $b(\gamma )$.

Denote by $\mu$ the push-forward of $\sigma |_{{\mathcal {G}}(K)}$ under the map $T_{K}$, and denote by $\mu _m^{}$ the push-forward of $\sigma _m|_{{\mathcal {G}}(K_m)}$ under the map $T_{m}$. By Lemma 7.4 and the Crofton formula,

$$\begin{aligned} \frac{1}{2} \cdot \mu _m^{}({\mathbb {R}}^2 \times {\mathbb {R}}^2)= & {} \text {Perimeter}_m(K_m) \xrightarrow {m \rightarrow \infty } \text {Perimeter}(K) \nonumber \\= & {} \frac{1}{2} \cdot \mu ({\mathbb {R}}^2 \times {\mathbb {R}}^2). \end{aligned}$$

(80)

For a convex polygon $P \subseteq {\mathbb {R}}^2$ we write ${\mathcal {F}}(P) \subseteq \partial K \times \partial K$ for the collection of all pairs of points $x \ne y \in \partial K$ with $(x,y) \cap P \ne \emptyset$. For a $d_m$-polygon P we denote by ${\mathcal {F}}_m(P) \subseteq \partial K_m \times \partial K_m$ the collection of all pairs of points $x \ne y \in \partial K_{m}$ with $(x,y)_m \cap P \ne \emptyset$. Note that if ${\overline{P}} \subseteq K_m$ then by the Crofton formula,

$$\begin{aligned} \frac{1}{2} \cdot \mu _m^{}({\mathcal {F}}_m(P)) = \text {Perimeter}_m(P). \end{aligned}$$

(81)

For a subset $A \subseteq {\mathbb {R}}^2 \times {\mathbb {R}}^2$ and $\varepsilon > 0$ we write ${\mathcal {N}}_{\varepsilon }(A) \subseteq {\mathbb {R}}^2 \times {\mathbb {R}}^2$ for the Euclidean $\varepsilon$-neighborhood, i.e., the collection of all $(x,y) \in {\mathbb {R}}^2 \times {\mathbb {R}}^2$ for which there exists $(z,w) \in A$ with $|x - z| < \varepsilon$ and $|y - w| < \varepsilon$.

Lemma 7.6

Fix two convex polygons $P_0, P_1 \subseteq {\mathbb {R}}^2$ with $\overline{P_0} \subseteq P_1$ and $\overline{P_1} \subseteq K$. For $i=0,1$ abbreviate ${\mathcal {F}}_i = {\mathcal {F}}(P_i)$. Then there exists $\varepsilon _0 > 0$ such that

$$\begin{aligned} \limsup _{m \rightarrow \infty } \mu _m^{}({\mathcal {N}}_{\varepsilon _0}({\mathcal {F}}_0)) \le \mu ({\mathcal {F}}_1) = 2 \cdot \mathrm{perimeter}(P_1). \end{aligned}$$

(82)

Furthermore, for any $0< \varepsilon < \varepsilon _0$,

$$\begin{aligned} \liminf _{m \rightarrow \infty } \mu _m^{}({\mathcal {N}}_{\varepsilon }({\mathcal {F}}_1)) \ge \mu ({\mathcal {F}}_0) = 2 \cdot \mathrm{perimeter}(P_0). \end{aligned}$$

(83)

Proof

Recall that $P^{(m)} \subseteq {\mathbb {R}}^2$ was defined to be the $d_m$-polygon with the same vertices as P, which is well-defined for a sufficiently large m. Write ${\mathcal {F}}_i^{(m)} = {\mathcal {F}}_m(P_i^{(m)})$. According to Lemma 7.3 there exist $\varepsilon _0 > 0$ and $m_0 \ge 1$ such that for any $m \ge m_0$,

$$\begin{aligned}&{\mathcal {N}}_{\varepsilon _0}({\mathcal {F}}_0) \cap (\partial K_m \times \partial K_m) \subseteq {\mathcal {F}}_1^{(m)} \qquad \text{ and } \qquad \nonumber \\&{\mathcal {N}}_{\varepsilon _0}({\mathcal {F}}_{0}^{(m)}) \cap (\partial K \times \partial K) \subseteq {\mathcal {F}}_1. \end{aligned}$$

(84)

By increasing $m_0$ if necessary, we may assume that $\overline{P_{i}^{(m)}} \subseteq K \subseteq K_{m}$ for all $m \ge m_0$ and $i=0,1$. Using (84) and (81), for $m \ge m_0$,

$$\begin{aligned} \mu _m^{}( {\mathcal {N}}_{\varepsilon _0}({\mathcal {F}}_0) ) \le \mu _m^{}( {\mathcal {F}}_1^{(m)} ) = 2 \cdot \text {Perimeter}_m(P_1^{(m)}) \xrightarrow {m \rightarrow \infty } 2 \cdot \text {Perimeter}(P_1). \end{aligned}$$

Fix $0< \varepsilon < \varepsilon _0$. By Lemma 7.4, there exists $m_1 \ge m_0$ such that for any $m \ge m_1$, the Hausdorff distance between $\partial K_m$ and $\partial K$ is at most $\varepsilon$. From (84) we obtain that for $m \ge m_1$,

$$\begin{aligned} {\mathcal {F}}_0^{(m)} \subseteq {\mathcal {N}}_{\varepsilon }({\mathcal {F}}_1). \end{aligned}$$

Hence,

$$\begin{aligned} \mu _m^{}( {\mathcal {N}}_{\varepsilon }({\mathcal {F}}_1) ) \ge \mu _m^{}( {\mathcal {F}}_0^{(m)} ) = 2 \cdot \text {Perimeter}_m(P_0^{(m)}) \xrightarrow {m \rightarrow \infty } 2 \cdot \text {Perimeter}(P_0). \end{aligned}$$

This completes the proof of (82) and (83). $\square$

Proof of Proposition 7.1

By passing to a subsequence, we may assume that $\text {Area}_m(D)$ converges to an element of ${\mathbb {R}}\cup \{ + \infty \}$ as $m \rightarrow \infty$, and our goal is to prove that this limit equals $\pi = \text {Area}(D)$.

The total mass of the measures $\mu _m^{}$ is uniformly bounded, by (80). Lemma 7.4 implies that the support of $\mu _m^{}$, which is contained in $\partial K_m \times \partial K_m$, is uniformly bounded in ${\mathbb {R}}^2$. We may thus pass to a subsequence and assume that

$$\begin{aligned} \mu _m^{} \xrightarrow {m \rightarrow \infty } {\tilde{\mu }} \end{aligned}$$

(85)

weakly for some measure ${\tilde{\mu }}$. This means that for any continuous test function $\varphi$ on ${\mathbb {R}}^2 \times {\mathbb {R}}^2$ we have $\int \varphi d \mu _m^{} \longrightarrow \int \varphi d {\tilde{\mu }}$. The measure ${\tilde{\mu }}$ is supported on $\partial K \times \partial K$, by Lemma 7.4.

Recall that $\overline{D(0,2)} \subseteq K$. We claim that for any convex polygon $P \subseteq {\mathbb {R}}^2$ with $P \subseteq D(0,2)$,

$$\begin{aligned} {\tilde{\mu }}({\mathcal {F}}(P)) = \mu ({\mathcal {F}}(P)). \end{aligned}$$

(86)

Since $\mu ({\mathcal {F}}(P)) = 2 \cdot \text {Perimeter}(P)$ is continuous in P and monotone in P under inclusion, and since ${\mathcal {F}}(P') \subseteq {\mathcal {F}}(P)$ when $P' \subseteq P$, in order to prove (86) it suffices to prove the following: For any two convex polygons $P_0, P_1 \subseteq {\mathbb {R}}^2$ with $\overline{P_0} \subseteq P_1$ and $\overline{P_1} \subseteq K$,

$$\begin{aligned} {\tilde{\mu }}({\mathcal {F}}(P_0)) \le \mu ({\mathcal {F}}(P_1)) \qquad \text {and} \qquad \mu ({\mathcal {F}}(P_0)) \le {\tilde{\mu }}(\overline{{\mathcal {F}}(P_1)}). \end{aligned}$$

(87)

From Lemmas 7.6 and (85), there exists $\varepsilon _0 > 0$ with

$$\begin{aligned} {\tilde{\mu }}({\mathcal {F}}(P_0)) \le \limsup _{m \rightarrow \infty } \mu _m({\mathcal {N}}_{\varepsilon _0}({\mathcal {F}}(P_0))) \le \mu ({\mathcal {F}}(P_1)) \end{aligned}$$

and for any $0< \varepsilon < \varepsilon _0$,

$$\begin{aligned} {\tilde{\mu }}({\mathcal {N}}_{2 \varepsilon }({\mathcal {F}}(P_1))) \ge \liminf _{m \rightarrow \infty } \mu _m( {\mathcal {N}}_{\varepsilon }({\mathcal {F}}(P_1)) ) \ge \mu ({\mathcal {F}}(P_0)). \end{aligned}$$

By letting $\varepsilon$ tend to zero, we obtain (87), and hence (86) is proven. The map $T_K^{-1}$ is a well-defined map from $A = \{ (x,y) \in \partial K \times \partial K \, ; \, x \ne y \}$ to ${\mathcal {G}}$. By (86), the push-forward of ${\tilde{\mu }}|_A$ under the map $T_K^{-1}$ is a measure ${\tilde{\sigma }}$ on ${\mathcal {G}}$ which satisfies the assumptions of Lemma 7.2. From the conclusion of Lemma 7.2,

$$\begin{aligned}&\frac{1}{2 \pi } \int _{\partial K \times \partial K} \text {Length}((x,y) \cap D ) d {\tilde{\mu }}(x,y) \nonumber \\&\quad = \frac{1}{2 \pi } \int _{A} \text {Length}((x,y) \cap D ) d {\tilde{\mu }}(x,y) = \pi . \end{aligned}$$

(88)

By the Santaló formula,

$$\begin{aligned} \text {Area}_m(D_m^{\pm }) = \frac{1}{2\pi } \int _{{\mathbb {R}}^2 \times {\mathbb {R}}^2} \text {Length}_m( (x,y)_m \cap D_m^{\pm } ) d \mu _m^{}(x,y). \end{aligned}$$

We thus deduce from (79), (85) and (88) that

$$\begin{aligned} \lim _{m \rightarrow \infty } \text {Area}_m(D_m^{\pm }) = \frac{1}{2\pi } \int _{{\mathbb {R}}^2 \times {\mathbb {R}}^2} \text {Length}( (x,y) \cap D) d {\tilde{\mu }}(x,y) = \pi . \end{aligned}$$

(89)

However, $D_m^- \subseteq D \subseteq D_m^+$. Hence (89) implies that $\text {Area}_m(D) \longrightarrow \pi$. $\square$