  The problem of classifying high-dimensional shapes in real-world data grows
in complexity as the dimension of the space increases. For the case of
identifying convex shapes of different geometries, a new classification
framework has recently been proposed in which the intersections of a set of
one-dimensional representations, called rays, with the boundaries of the shape
are used to identify the specific geometry. This ray-based classification (RBC)
has been empirically verified using a synthetic dataset of two- and
three-dimensional shapes (Zwolak et al. in Proceedings of Third Workshop on
Machine Learning and the Physical Sciences (NeurIPS 2020), Vancouver, Canada
[December 11, 2020], arXiv:2010.00500, 2020) and, more recently, has also been
validated experimentally (Zwolak et al., PRX Quantum 2:020335, 2021). Here, we
establish a bound on the number of rays necessary for shape classification,
defined by key angular metrics, for arbitrary convex shapes. For two
dimensions, we derive a lower bound on the number of rays in terms of the
shape's length, diameter, and exterior angles. For convex polytopes in
$\mathbb{R}^N$, we generalize this result to a similar bound given as a
function of the dihedral angle and the geometrical parameters of polygonal
faces. This result enables a different approach for estimating high-dimensional
shapes using substantially fewer data elements than volumetric or surface-based
approaches.
