The problem
A ball travels along straight lines inside a circular table and bounces with equal incoming and outgoing angles. If its path starts tangent to a smaller concentric circle, every later segment stays tangent to that same circle. Different starting angles give different inner circles.
Could a smooth, strictly convex table have such smoothly varying inner curves filling an entire band near its wall, yet fail to be an ellipse? Circles count as ellipses. The unknown requires a full band of these curves, not just some special trajectories or isolated inner curves.[1] [2]
Could a table that is not an ellipse still support smoothly varying inner curves filling a whole band near its wall, with every tangent billiard path staying tangent after all bounces? The ball moves freely in straight lines and reflects with equal incoming and outgoing angles. The table is planar, smooth and strictly convex.
A circle gives a concrete example of the behavior. In a unit circular table, begin at with unit direction . The straight path is at distance from the center, so it is tangent to the concentric circle of that radius. Its next boundary point is ; reflection there gives unit direction . The reflected line is again at distance from the center.
Such a preserved smooth convex inner curve is a caustic. A smooth foliation requires these noncrossing curves to vary smoothly and fill an open band near the boundary, corresponding to invariant curves for near-tangent billiard states. The target is this full near-boundary family, not a foliation of the entire phase space and not merely some caustics accumulating at the wall. Must the table be an ellipse, including the circular case?[1] [2]
Scope
The near-boundary formulation: smooth strictly convex planar tables with a smooth foliation by convex caustics near their boundary should be ellipses, including circles.
What remains open
Results near ellipses give part of the classification: under specified conditions on repeating billiard paths tangent to the inner curves, a sufficiently small deformation of an ellipse must itself be an ellipse. Koval’s result applies for almost every ellipse, with excluded eccentricities and explicit closeness and smoothness requirements. These local results leave tables far from an ellipse uncovered, and do not classify every smooth strictly convex table with the whole near-wall band in the question.[1] [2]
Koval proves a local rigidity result for sufficiently small perturbations of almost every ellipse, under a specified near-boundary rational-integrability condition. The exceptional eccentricities and closeness assumptions remain part of that theorem. The earlier Kaloshin–Sorrentino work is likewise local near ellipses.
These results do not classify arbitrary smooth strictly convex tables with the full caustic foliation in this page’s scope. A collection of caustics accumulating near a boundary, or integrability near one periodic orbit, does not by itself supply that full foliation.[1] [2]
Why it matters
The bounce rule is local: it uses the wall at the point of contact. Remaining tangent to one inner curve is a constraint on the whole future path. Connecting those two features turns motion into evidence about shape, making billiards an inverse problem rather than only a study of paths on a known table.[1] [2]
A trajectory state includes both its point of contact and its direction. A caustic organizes many such states into an invariant family, giving information that survives repeated reflections rather than one locally correct bounce. In the circle, the fixed distance of each tangent line from the center is one visible instance of that persistence.
Recovering a boundary from these invariant families is an inverse-dynamics question. The cited work also connects billiard paths and their lengths with spectral questions about domains. That is a methodological link to geometric inverse problems, not a claim that the full conjecture or a general shape-recovery algorithm has been proved.[1] [2]
Definitions
3 terms
caustic
A smooth convex curve inside the table such that any billiard path tangent to it remains tangent after every reflection. It is a preserved envelope for paths, not the path of a single ball.[1] [2]
smooth foliation
Here, a smoothly varying family of noncrossing convex caustics filling an open band near the boundary. A collection with gaps or just some caustics accumulating at the wall is weaker.[1] [2]
strictly convex
A convex table whose boundary has no straight segment: the open segment between any two distinct boundary points lies inside the table. This geometric condition alone need not give positive curvature at every point; the cited local theorems retain their own smoothness and curvature hypotheses.[2]
Sources
- Local strong Birkhoff conjecture and local spectral uniqueness of almost every ellipse
Notes on Local strong Birkhoff conjecture and local spectral uniqueness of almost every ellipse
Introduction defines near-boundary integrability and proves a perturbative result.
- On the Local Birkhoff Conjecture for Convex Billiards
Notes on On the Local Birkhoff Conjecture for Convex Billiards
Introduction separates near-boundary caustics from stronger global foliation.