Let $\Omega \subset \mathbf{R}^2$ be a bounded, simply connected domain, with a regular boundary, say of class $C^2$ at least. Let the cut locus $C$ of $\Omega$ be the set of points $x \in \Omega$ for which there exist two or more points $z_1,z_2 \in \partial \Omega$ for which \begin{equation} \lvert x - z_i \rvert = \operatorname{dist}(x,\partial \Omega). \end{equation}

It is claimed in a paper of Panov and Petrunin that an arbitrarily small perturbation of $\gamma$ guarantees that the cut locus $C$ is a finite graph, embedded inside $\Omega$. (In fact, once the graph structure of $C$ is established one can show that $C$ is a tree.) They attribute this fact to Ionin and Pestov, but unfortunately this is available only in Russian.

Question. How does this perturbation argument go? (And for which 'pathological' domains is it necessary in the first place?)

Let $\Omega \subset \mathbf{R}^2$ be a bounded, simply connected domain, with a regular boundary, say of class $C^2$ at least. Let the cut locus $C$ of $\Omega$ be the set of points $x \in \Omega$ for which there exist two or more points $z_1,z_2 \in \partial \Omega$ for which \begin{equation} \lvert x - z_i \rvert = \operatorname{dist}(x,\partial \Omega). \end{equation}

It is claimed in a paper of Panov and Petrunin that an arbitrarily small perturbation of the boundary guarantees that the cut locus $C$ is a finite graph, embedded inside $\Omega$. (In fact, once the graph structure of $C$ is established one can show that $C$ is a tree.) They attribute this fact to Ionin and Pestov, but unfortunately this is available only in Russian.

Question. How does this perturbation argument go? (And for which 'pathological' domains is it necessary in the first place?)

Let $\gamma: \mathbf{S}^1 \to \mathbf{R}^2$ be a simple, closed curve. Assume additionally that $\gamma$ is regular, of class $C^2$ at least. The complement of $\gamma$ in $\mathbf{R}^2$ has two connected components, of which we denote $\Omega \subset \mathbf{R}^2$ that which is bounded.

We define the cut locus of $\Omega$ to be the closure of the set of points $x \in \Omega$ for which there exist two or more points $z_1,z_2 \in \partial \Omega = \gamma$ with $\lvert x-z_i \rvert = \mathrm{dist}(x,\Omega)$. We denote this set $C \subset \Omega$. (This coincides with the definition of the cut locus of $\partial \Omega$ in terms of the exponential map, when endowing $\Omega$ with the Euclidean metric.)

It is claimed in a paper of Panov and Petrunin that an arbitrarily small perturbation of $\gamma$ guarantees that the cut locus $C$ is a finite graph, embedded inside $\Omega$. In fact, once the graph structure of $C$ is established one can show that $C$ is a tree. The authors seem to attribute these observations to Ionin and Pestov, but unfortunately this latter paper is available only in Russian.

Question: What is the formal justification of this perturbation argument?

Let $\Omega \subset \mathbf{R}^2$ be a bounded, simply connected domain, with a regular boundary, say of class $C^2$ at least. Let the cut locus $C$ of $\Omega$ be the set of points $x \in \Omega$ for which there exist two or more points $z_1,z_2 \in \partial \Omega$ for which \begin{equation} \lvert x - z_i \rvert = \operatorname{dist}(x,\partial \Omega). \end{equation}

It is claimed in a paper of Panov and Petrunin that an arbitrarily small perturbation of $\gamma$ guarantees that the cut locus $C$ is a finite graph, embedded inside $\Omega$. (In fact, once the graph structure of $C$ is established one can show that $C$ is a tree.) They attribute this fact to Ionin and Pestov, but unfortunately this is available only in Russian.

Question. How does this perturbation argument go? (And for which 'pathological' domains is it necessary in the first place?)

Let $\gamma: \mathbf{S}^1 \to \mathbf{R}^2$ be a simple, closed curve. Assume additionally that $\gamma$ is regular, of class $C^2$ at least. The complement of $\gamma$ in $\mathbf{R}^2$ has two connected components, of which we denote $\Omega \subset \mathbf{R}^2$ that which is bounded. (This has boundary $\partial \Omega = \gamma$.)

We define the cut locus of $\Omega$ to be the closure of the set of points $x \in \Omega$ for which there exist two or more points $z_1,z_2 \in \partial \Omega$ with $\lvert x-z_i \rvert = \mathrm{dist}(x,\Omega)$. We denote this set $C \subset \Omega$. (This coincides with the definition of the cut locus of $\partial \Omega$ in terms of the exponential map, when endowing $\Omega$ with the Euclidean metric.)

It is claimed in a paper of Panov and Petrunin that an arbitrarily small perturbation of $\gamma$ guarantees that the cut locus $C$ is a finite graph, embedded inside $\Omega$. In fact, once the graph structure of $C$ is established one can show that $C$ is a tree. The authors seem to attribute these observations to Ionin and Pestov, but unfortunately this latter paper looks to be available only in Russian.

Question: What is the formal justification of this perturbation argument? It looks like Sard's theorem might apply, but then that is mainly concerned with regular sets, which $C$ is not. What `pathological' shapes of $\gamma$ lead to a cut locus $C$ which is not a finite graph?

Let $\gamma: \mathbf{S}^1 \to \mathbf{R}^2$ be a simple, closed curve. Assume additionally that $\gamma$ is regular, of class $C^2$ at least. The complement of $\gamma$ in $\mathbf{R}^2$ has two connected components, of which we denote $\Omega \subset \mathbf{R}^2$ that which is bounded.

We define the cut locus of $\Omega$ to be the closure of the set of points $x \in \Omega$ for which there exist two or more points $z_1,z_2 \in \partial \Omega = \gamma$ with $\lvert x-z_i \rvert = \mathrm{dist}(x,\Omega)$. We denote this set $C \subset \Omega$. (This coincides with the definition of the cut locus of $\partial \Omega$ in terms of the exponential map, when endowing $\Omega$ with the Euclidean metric.)

It is claimed in a paper of Panov and Petrunin that an arbitrarily small perturbation of $\gamma$ guarantees that the cut locus $C$ is a finite graph, embedded inside $\Omega$. In fact, once the graph structure of $C$ is established one can show that $C$ is a tree. The authors seem to attribute these observations to Ionin and Pestov, but unfortunately this latter paper is available only in Russian.

Question: What is the formal justification of this perturbation argument?