Baker's Dozen — Periodic multidimensional outer-billiard trajectories

Definitions and mathematical statement

Let $n\geq1$ be an integer, and let $M\subset{\mathbb{R}}^{2n}$ be a smooth strictly convex hypersurface. The source asks for lower bounds on the number of $p$-periodic outer-billiard orbits for integer periods $p\ne3$.

To describe the geometric setting, equip ${\mathbb{R}}^{2n}$ with its standard symplectic form $\omega=\sum_{j=1}^{n}{\,\mathrm{d}} q_j\wedge{\,\mathrm{d}} r_j$. The characteristic direction at $m\in M$ is the one-dimensional kernel of $\omega$ restricted to $T_mM$. The usual geometric outer-billiard construction reflects a point $z$ through a tangency point $m\in M$ along this direction, so that the next point is $2m-z$. Defining a single-valued outer-billiard map $T_M$ requires the appropriate oriented tangency convention and domain. These are not specified in the supplied record. For a chosen such map, a periodic trajectory satisfies \[ z_{i+1}=T_M(z_i),\qquad i\in{\mathbb{Z}}/p{\mathbb{Z}}. \] The problem is to obtain lower bounds for the number of such orbits, for positive integer $p\ne3$, in the stated smooth strictly convex class.

The record does not specify whether $M$ must be closed and bound a compact convex body, whether "$p$-periodic" means least period exactly $p$ or period dividing $p$, or which identifications define distinct orbits. It also does not specify whether the intended bounds are uniform in $M$, their dependence on $n,p$, or any nondegeneracy assumption. These are material qualifications to resolve before a unique numerical counting assertion can be formulated. No stronger curvature or genericity assumption is imposed here.

Short English statement

For multidimensional outer billiards outside a smooth strictly convex hypersurface, find guarantees on the number of periodic trajectories for periods other than three. The supplied 2015 list presents the problem as open; the registry later labels it partially solved, with current status still requiring review.

The supplied research summary proposes a variational formulation on the cyclic configuration space \[ \{(x_1,\ldots,x_p)\in(S^{2n-1})^p:x_i\ne x_{i+1}\},\qquad x_{p+1}=x_1, \] using a generating function $h$ and $F=\sum_i h(x_i,x_{i+1})$. It suggests that cohomology and Morse--Lusternik--Schnirelmann methods could yield a bound of the shape $(p-1)(2n-2)+1$ for prime $p$, conditional on resolving the compactness issue near collision diagonals. This is a proposed route in the supplied summary, not an established lower bound in this draft. The summary is truncated mid-sentence, so its full hypotheses and conclusion are unavailable; its assertion that strict convexity identifies $M$ with a sphere via the Gauss map also requires clarification of the regularity and curvature conventions.

Sources

• UnsolvedMath permanent ID 600006, source code catalogue entry, "Baker's Dozen --- Periodic multidimensional outer-billiard trajectories." Registry curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0; record updated 2026-08-06.
• Tabachnikov, A Baker's Dozen of Problems (2015), Problem 2, as identified by the supplied AMR-indexed list: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/. Publisher HTML extraction accessed 2026-07-29. The record marks status evidence and rights review as NEEDS_REVIEW.
• The supplied, truncated research summary attributes a cyclic-configuration-space cohomology computation to Farber--Tabachnikov (2002). It supplies no complete bibliographic reference or locator for that claim; no further citation is inferred.