Geometric optimization problems generated by plane curves
Let $γ_1$ and $ γ_2$ be regular $C^1$-smooth curves in the plane and $γ$ be a regular $C^2$-smooth curve in the same plane. Consider all triples of points $(A, A_1, A_2)$, $A\in γ$, $A_1\in γ_1$, $A_2\in γ_2$, such that $A_1\neq A_2$, and the line $A_1 A_2$ is the normal to $γ$ at $A$. We show that, if $γ$ has non-vanishing curvature and the triple $(A^0, A_1^0,A_2^0 )$ is a local maximum or a local minimum for the distance $|A_1A_2|$ between the points $A_1$ and $A_2$, then the following three lines either meet at a single point or are parallel: the normal to $γ_1$ at $A_1^0$, the normal to $γ_2$ at $A_2^0$ and the line which is perpendicular to $A_1^0A_2^0$, and passing through the center of curvature of $γ$ at $A^0$. The particular case of this optimization problem, when $γ$ is a circle with a given center $O$, coincides with the already partially studied problem of finding the locally shortest (or the locally longest) non-degenerate segments $[A_1A_2]$ such that $A_1\inγ_1$, $A_2\inγ_2$ and $O\in A_1A_2$. We also show that the seemingly different problem of finding the locally shortest (or the locally longest) non-degenerate segments $[A_1A_2]$, such that $A_1 \in γ_1$, $A_2 \in γ_2$, and the line $A_1A_2$ is tangent to $γ$ is also, in essence, a particular case of the above optimization problem. We consider in detail the ``degenerate cases'' naturally appearing in this setting (when, for instance, $γ_1$ or $γ_2$ coincide with $γ$, or when the optimal line $A_1^0A_2^0$ is tangent to at least one of $γ_1$ or $γ_2$).