Consider a convex body $\Omega\subset \mathbb{R}^2$. Let $L(d)$ be the maximum over all curves $C$ of degree $d$ of the length of $C\cap\Omega$.

Is $L(d)\leq d P(E)/2$, where $P(E)$ is the perimeter of the longest ellipse inscribed in $\Omega$?

It is clear that $L(d)\geq dP(E)/2$ if $d$ is even. If $\Omega$ is a ball, then the matching upper bound follows from Buffon's noodle argument. In general, one has $L(d)\leq dP(\Omega)/2$, where $P(\Omega)$ is the perimeter of $\Omega$.

Is the answer for $\Omega=[-1,+1]^2$ really the same as $\Omega=B(0,1)$? Another attractive question is for equilateral triangle (see comments of Noam Elkies below).

In the answer below, jacob hints on an example that is likely to be a counterexample to the question above, namely, convex hull of the graph of $\{Tx^2 : x \in [-1,+1]\}$. Perhaps the right question is whether $L(d)$ is maximized by a quadratic curve.

compactconvex body? $\endgroup$ – David Roberts Dec 17 '13 at 2:37convex bodymeans a compact convex set with non-empty interior. $\endgroup$ – Boris Bukh Dec 17 '13 at 2:49