research/AMR-005-0011.md

---
id: AMR-005-0011
classification: PARTIAL-PROGRESS
wording_corrected: no
---
# AMR-005-0011 — Convex tangent-segment iteration (bicycle front-track map)

## Problem (corrected statement if needed)

The transcription was checked against the source (S. Tabachnikov, "A Baker's Dozen
of Problems", Arnold Math. J. 1 (2015), §9, Conjecture 4) and is accurate; no
correction needed. Original statement:

> Given an oriented oval $\gamma$, draw the unit tangent segments to $\gamma$, and
> let $\gamma_1$ be the locus of their endpoints. We get a map
> $\gamma\mapsto\gamma_1$. **Conjecture:** If all iterations of this map are convex
> curves then $\gamma$ is a circle.

Equivalently: parametrize the oval by its tangent angle $\theta$,
$X'(\theta)=\rho(\theta)(\cos\theta,\sin\theta)$ with radius of curvature
$\rho>0$ and closure condition $\int_0^{2\pi}\rho\,e^{i\theta}\,d\theta=0$.
The map is $Y(\theta)=X(\theta)+(\cos\theta,\sin\theta)$ (in bicycle language:
$\gamma$ is the rear-wheel track, $\gamma_1$ the front-wheel track of a unit
segment). Conjecture: the only ovals whose whole forward orbit stays convex are
circles. Circles do survive: $\rho\equiv R\Rightarrow \rho_1\equiv\sqrt{1+R^2}$.

## Status / Literature

- Source: S. Tabachnikov, *A Baker's Dozen of Problems*, Arnold Math. J. 1 (2015),
  §9, Conjecture 4 (verified at the journal page:
  https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/).
  Tabachnikov's justification: the linearized statement is a theorem — if $F$ is
  periodic and all iterates of $F\mapsto F+F'$ are positive, then $F$ is a
  positive constant.
- Related literature on the same map (bicycle/unicycle kinematics):
  M. Levi & S. Tabachnikov, *On bicycle tire tracks geometry, hatchet planimeter,
  Menzin's conjecture and oscillation of unicycle tracks*, Exp. Math. 18 (2009)
  173–186 — for the open-arc ("unicycle") version they prove the number of local
  extrema and of inflection points strictly increases at each iteration, i.e.
  complexity grows under the same map;
  R. Foote, M. Levi, S. Tabachnikov, *Tractrices, bicycle tire tracks, hatchet
  planimeters, and a 100-year-old conjecture*, Amer. Math. Monthly 120 (2013)
  199–216;
  G. Bor, M. Levi, R. Perline, S. Tabachnikov, *Tire tracks and integrable curve
  evolution*, arXiv:1705.06314 (IMRN 2021) — relates the continuous bicycle flow
  to the AKNS system and the filament equation (checked the abstract; it does not
  address this convexity conjecture).
- I found **no published resolution** of Conjecture 4. Caveat: web search was
  heavily rate-limited during this work (repeated HTTP 429 from the search
  provider and arXiv/Semantic Scholar APIs); the negative finding is based on
  the source article, the arXiv abstract of arXiv:1705.06314, and one successful
  DuckDuckGo query, so a very recent resolution cannot be fully excluded.

## Work done

All derivations below are mine and were checked numerically.

**1. Exact convexity criterion and curvature evolution.** With the tangent-angle
parametrization,
$$Y'(\theta)=(\rho\cos\theta-\sin\theta,\ \rho\sin\theta+\cos\theta),\quad
|Y'|=\sqrt{1+\rho^2}>0,$$
so the tangent angle of $\gamma_1$ is
$\varphi(\theta)=\theta+\arctan(1/\rho(\theta))$, and its radius of curvature is
$$\rho_1(\varphi)=\frac{(1+\rho^2)^{3/2}}{\,1+\rho^2-\rho'\,},
\qquad \varphi'=\frac{1+\rho^2-\rho'}{1+\rho^2}.$$
Since a regular closed curve with strictly monotone tangent angle of total
increment $2\pi$ is convex, we get the sharp criterion
$$\gamma_1\ \text{convex}\ \iff\ 1+\rho(\theta)^2-\rho'(\theta)>0\ \ \forall\theta.$$
The conjecture is thus equivalent to: *the only positive periodic $\rho$ with
$\int\rho e^{i\theta}=0$ whose whole forward orbit under
$\rho\mapsto(1+\rho^2)^{3/2}/(1+\rho^2-\rho')$ (with the reparametrization
$\varphi=\theta+\arctan(1/\rho)$) keeps satisfying $1+\rho_n^2-\rho_n'>0$ is
$\rho\equiv\mathrm{const}$.*

**2. Rigorous proof of Tabachnikov's linearized model.** Claim: if $F$ is a
smooth $2\pi$-periodic real function and $T^nF>0$ for all $n\ge0$, where
$TF=F+F'$, then $F$ is a positive constant. Proof: $\widehat{T^nF}(k)=(1+ik)^n\hat
F(k)$ and the mean of $T^nF$ is $\hat F(0)$ for all $n$. If $\hat F(m)\neq0$ for
some $m\neq0$, then
$\sup(T^nF)-\inf(T^nF)\ge 2(1+m^2)^{n/2}|\hat F(m)|\to\infty$ while the mean
stays $\hat F(0)$, so $T^nF$ takes negative values for large $n$ — contradiction.
Hence $F\equiv\hat F(0)>0$. $\blacksquare$

**3. Linearized instability of the circle (rigorous).** Writing
$\rho=R+\varepsilon u(\theta)$ and linearizing the map of part 1:
$$\rho_1(\varphi)=\sqrt{1+R^2}+\frac{\varepsilon}{\sqrt{1+R^2}}
\big(Ru(\theta)+u'(\theta)\big)+O(\varepsilon^2),\qquad
\varphi=\theta+c-\tfrac{\varepsilon u}{1+R^2}+O(\varepsilon^2),\ c=\arctan(1/R).$$
The linearized operator $L_Ru=\big(Ru(\cdot-c)+u'(\cdot-c)\big)/\sqrt{1+R^2}$ is
diagonalized by $e^{ik\theta}$ with eigenvalues
$$\lambda_k(R)=\frac{(R+ik)e^{-ikc}}{\sqrt{1+R^2}},\qquad
|\lambda_k|=\sqrt{\frac{R^2+k^2}{R^2+1}}.$$
$k=\pm1$ (the closure/translation modes) are neutral, $k=0$ is damped, and every
mode $|k|\ge2$ is **amplified**: $|\lambda_k|>1$. Since $R_n=\sqrt{R_0^2+n}$ for
the circle orbit, the cumulative amplification of mode $k$ over $n$ steps is
$\prod_j\sqrt{(R_j^2+k^2)/(R_j^2+1)}\asymp n^{(k^2-1)/2}$, while convexity
survival needs $\rho_n'<1+\rho_n^2\sim n$; the linear estimate predicts breakup
around $n\sim\varepsilon^{-2}$ for mode 2 (nonlinear breakup is observed much
earlier). So circles are linearly unstable fixed points of the rescaled dynamics
— the conjecture is the statement that this instability is never tamed
nonlinearly.

**4. Numerical experiment** (spectral code, `/tmp/amr_iter.py`,
`/tmp/amr_iter4.py`; numpy/scipy in an isolated venv; grids $N=8192$–$131072$,
spectral derivative, periodic cubic-spline resampling onto the uniform
$\varphi$-grid, closure projection, low-pass filter):

- Circle $\rho\equiv1$: iterates match $R_n=\sqrt{n+1}$ and
  $\min(1+\rho^2-\rho')=1+R_n^2$ to machine precision over 30 steps (validates
  the code).
- Early growth of a mode-2 perturbation matches the linear prediction
  $|\lambda_2(R)|=\sqrt{(R^2+4)/(R^2+1)}$ step by step (e.g. absolute amplitude
  $\times1.581$ at $R=1$, $\times1.414$ at $R=\sqrt2$).
- Every tested non-circular oval lost convexity within a few steps:
  $\rho=1+0.2\cos2\theta$: lost at $n=4$; $1+0.05\cos2\theta$ and
  $1+0.01\cos2\theta$: lost at $n=6$; $1+0.2\cos3\theta$: lost at $n=2$;
  $1+0.05\cos4\theta$: lost at $n=1$; $2+0.3\cos2\theta$: lost at $n=3$.
  The loss is abrupt: $\min(1+\rho^2-\rho')$ jumps from $\approx4$ to
  $\approx-10^2$ in one step — the denominator $1+\rho^2-\rho'$ dips, $\rho_1$
  spikes locally, and the next derivative explodes (a nonlinear sharpening
  cascade, consistent with part 3's amplification of high modes).
- Honest caveat: for very small amplitudes ($\le10^{-3}$) the *step* of breakdown
  shifts with grid resolution (loss at $n=5$–$6$ depending on $N$), because
  resampling noise in high Fourier modes is amplified $\sim k$ per step by the
  map; the quoted growth rates and the moderate-amplitude phenomenology are
  resolution-stable, but the exact lifetime of tiny perturbations is not
  numerically trustworthy.

## Result

No full solution. Rigorous partial progress: (i) exact reformulation as a
1-dimensional curvature dynamical system with the sharp convexity criterion
$1+\rho^2-\rho'>0$; (ii) complete proof of the linearized model $F\mapsto F+F'$;
(iii) rigorous linear analysis showing circles are isolated, linearly unstable
fixed points of the rescaled dynamics, with all non-trivial Fourier modes
($|k|\ge2$) amplified by $\sqrt{(R^2+k^2)/(R^2+1)}$ per step; (iv) numerical
evidence that generic non-circular ovals lose convexity within a handful of
iterations via a sharpening cascade. Combined with Levi–Tabachnikov's theorem
that complexity (numbers of extrema/inflections) strictly increases under the
same map for open arcs, the conjecture is very plausible but, to my knowledge,
still open.

## What remains

- A proof (or counterexample) of the conjecture. Natural routes: (a) find a
  monotone quantity (e.g. a weighted $L^2$ norm of the nonconstant Fourier part
  of $\rho$, or a geometric functional like isoperimetric defect normalized by
  scale) that strictly increases unless $\rho$ is constant; (b) promote the
  linear analysis to a nonlinear instability statement (invariant-manifold
  argument around the circle orbit $R_n=\sqrt{R_0^2+n}$); (c) use the
  integrability machinery of Bor–Levi–Perline–Tabachnikov (the map is one step
  of the discrete bicycle flow, tied to the discrete mKdV/AKNS hierarchy).
- A definitive numerical lifetime law $n_*(\varepsilon,k)$ for small
  perturbations requires noise-free high-precision numerics (spectral regridding
  without interpolation, or extended precision), which I did not complete.
- A more thorough literature pass once search-rate limits lift (the negative
  literature finding here is not exhaustive).