Baker's Dozen — Convex tangent-segment iteration

Definitions and mathematical statement

An oriented oval is a smooth embedded closed curve $\gamma:S^1\to\mathbb R^2$ bounding a strictly convex region. For a regular parametrization define \[ \tau_\gamma(t)=\frac{\gamma'(t)}{\|\gamma'(t)\|},\qquad T\gamma(t)=\gamma(t)+\tau_\gamma(t). \] The image $T\gamma$ is the locus of the endpoints of oriented unit tangent segments. Reparametrizations preserving orientation give the same geometric curve. When iterating $T$, compute the unit tangent of the new curve each time; one cannot keep the original speed parameter and replace all later unit tangents by unnormalized derivatives.

Put $\gamma_0=\gamma$ and $\gamma_{j+1}=T\gamma_j$ for $j\ge0$. The rigidity conjecture states that if every $\gamma_j$ remains an embedded convex closed curve, then the initial curve is a Euclidean circle. Convexity concerns the entire image and rules out folds and self-intersections.

The conclusion is up to translation, rotation, and the initial radius. Circles do satisfy the hypothesis: the tangent offset of a circle of radius $R$ is a circle of radius $\sqrt{R^2+1}$, with an angular shift. The problem is whether any noncircular oval can retain convexity indefinitely.

Short English statement

Does indefinite preservation of convexity under unit tangent offsets characterize circles among smooth convex closed plane curves?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 600011 (AMR-005-0011), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Tabachnikov - A Baker's Dozen of Problems (2015), Conjecture 4. Original source indexed by AMR-005-0011. https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/.