research/AMR-005-0011.md

metadata

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, $$Y^{\mathrm{\prime }}(\theta )=(\rho \cos \theta -\sin \theta ,\rho \sin \theta +\cos \theta ),\mathrm{\mid }{Y}^{\mathrm{\prime }}\mathrm{\mid }=\sqrt{1+{\rho }^2}>0,Y\text{'}(\backslash theta)=(\backslash rho\backslash cos\backslash theta-\backslash sin\backslash theta,\backslash \; \backslash rho\backslash sin\backslash theta+\backslash cos\backslash theta),\backslash quad\; |Y\text{'}|=\backslash sqrt\{1+\backslash rho^2\}>0,$$ so the tangent angle of $\gamma_1$ is $\varphi(\theta)=\theta+\arctan(1/\rho(\theta))$, and its radius of curvature is $${\rho }_1(\phi )=\frac{(1+{\rho }^2{)}^{3\mathrm{/}2}}{1+{\rho }^2-{\rho }^{\mathrm{\prime }}},{\phi }^{\mathrm{\prime }}=\frac{1+{\rho }^2-{\rho }^{\mathrm{\prime }}}{1+{\rho }^2}\mathrm{.}\backslash rho\_1(\backslash varphi)=\backslash frac\{(1+\backslash rho^2)^\{3/2\}\}\{\backslash ,1+\backslash rho^2-\backslash rho\text{'}\backslash ,\},\; \backslash qquad\; \backslash varphi\text{'}=\backslash frac\{1+\backslash rho^2-\backslash rho\text{'}\}\{1+\backslash rho^2\}.$$ Since a regular closed curve with strictly monotone tangent angle of total increment $2\pi$ is convex, we get the sharp criterion $${\gamma }_1\text{ convex }⟺1+\rho (\theta {)}^2-{\rho }^{\mathrm{\prime }}(\theta )>0\mathrm{\forall }\theta \mathrm{.}\backslash gamma\_1\backslash \; \backslash text\{convex\}\backslash \; \backslash iff\backslash \; 1+\backslash rho(\backslash theta)^2-\backslash rho\text{'}(\backslash theta)>0\backslash \; \backslash \; \backslash forall\backslash theta.$$ 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: $${\rho }_1(\phi )=\sqrt{1+R^2}+\frac{\epsilon }{\sqrt{1+R^2}}(Ru(\theta )+u^{\mathrm{\prime }}(\theta ))+O({\epsilon }^2),\phi =\theta +c-\frac{\epsilon u}{1+R^2}+O({\epsilon }^2),c=\arctan (1\mathrm{/}R)\mathrm{.}\backslash rho\_1(\backslash varphi)=\backslash sqrt\{1+R^2\}+\backslash frac\{\backslash varepsilon\}\{\backslash sqrt\{1+R^2\}\}\; \backslash big(Ru(\backslash theta)+u\text{'}(\backslash theta)\backslash big)+O(\backslash varepsilon^2),\backslash qquad\; \backslash varphi=\backslash theta+c-\backslash tfrac\{\backslash varepsilon\; u\}\{1+R^2\}+O(\backslash varepsilon^2),\backslash \; c=\backslash arctan(1/R).$$ 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 $${\lambda }_k(R)=\frac{(R+ik)e^{-ikc}}{\sqrt{1+R^2}},\mathrm{\mid }{\lambda }_k\mathrm{\mid }=\sqrt{\frac{R^2+k^2}{R^2+1}}\mathrm{.}\backslash lambda\_k(R)=\backslash frac\{(R+ik)e^\{-ikc\}\}\{\backslash sqrt\{1+R^2\}\},\backslash qquad\; |\backslash lambda\_k|=\backslash sqrt\{\backslash frac\{R^2+k^2\}\{R^2+1\}\}.$$ $k=\pm1$ (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).