Assume that there are two (competing) pizza houses situated at the points $0$ and $1$ on the complex plane. These pizza houses can deliver pizza to points of the plane with the largest velocities $v_0$ and $v_1$, respectively.

**Definition.** A closed subset $P$ of the complex plane is called a *$(v_0,v_1)$-pizza curve* if $P$ is a common boundary of two open connected sets $U_0,U_1$ in $\mathbb C$ such that

$0\in U_0$ and $1\in U_1$;

$U_0\cap U_1=\emptyset$ and $U_0\cup P\cup U_1=\mathbb C$.

for any point $z\in P$ there exists a positive real number $t_z$ such that

(i) for every $\varepsilon>0$ and every $k\in\{0,1\}$ there exists a smooth curve $\gamma_k:[0,t_z+\varepsilon)\to U_k$ such that $\gamma_k(0)=k$, $\lim_{t\to t_z+\varepsilon}\gamma_k(t)=z$ and $\lvert\gamma_k'(t)\rvert\le v_k$ for every $t\in[0,t_z+\varepsilon)$;

(ii) for every $\varepsilon>0$ and every $k\in\{0,1\}$ there exists no smooth curve $\gamma_0:[0,t_z-\varepsilon)\to U_k$ such that $\gamma_k(0)=k$, $\lim_{t\to t_z-\varepsilon}\gamma_k(t)=z$ and $\lvert\gamma_k'(t)\rvert\le v_k$ for every $t\in[0,t_z-\varepsilon)$.

Problem.What is the form of a $(v_0,v_1)$-pizza curve? Is it unique?

**Remark 1.** If $v_0=v_1$, then the answer to this problem is well-known: the $(v_0,v_1)$-pizza curve is unique and coincides with the line $\{z\in\mathbb C:\Re(z)=\frac12\}$. So the problem essentially concerns the case $v_0\ne v_1$.

**Remark 2.** It can be shown that each $(v_0,v_1)$-pizza curve locally coincides with the graph of some Lipschitz function.