Previous theoretical work has shown that, in an unbounded domain, anticyclones are prohibited from splitting on their own due to limitations imposed by the conservation of angular momentum. By explicitly considering the role of angular momentum exchange between eddies and boundaries (neglected by previous theories), splitting criteria for an anticyclonic lens colliding with a long and thin island are established analytically. The inviscid analytical model consists of an isolated patch of fluid in a reduced gravity regime. Nonlinear analytical solutions are constructed by connecting the initial and final states using conserved quantities (integrated angular momentum, vorticity and mass) and the familiar slowly varying approximation. For the conceptual case of a lens pierced by a thin moving wall, the result is that, in order for a zero potential vorticity lens (with a radius R1) to split into two equal offspring, the wall length must be at least 1.19 R1. Even for infinitesimal splitting, which arises from weak collisions (where the wall merely "brushes" the lens), the wall must be O (R1). This is because the "parent" lens can split into two offspring only when the wall allows sufficient spreading of the lens, which increases its relative angular momentum, and thereby enables the lens to form two distinct "offspring.'' Numerical experiments employing Lagrangian floats reveal that the splitting is accomplished by a jet that leaks fluid along the wall, forming a second lens. The fluid initially found along the rim of the parent lens occupies both the core and the rim of the second lens; the fluid found at an intermediate radius in the second lens is derived from fluid situated at an intermediate radius in the parent lens. In general, a very good agreement between the numerics and the analytical theory is found. The numerical simulations demonstrate that the integrated angular momentum is a far stronger constraint than energy conservation. Using the numerics, we extend the moving wall theory to the splitting of finite vorticity lenses and lenses on a β-plane. We find that the basic requirement of mass redistribution by a wall is relevant in all the regimes that we examined, and, therefore, is likely to also be relevant to collisions of eddies with actual islands. This supports our application of the theory to Meddy splitting by seamounts, where we find that the seamounts can provide the necessary torque for the recently observed Meddy splitting and destruction.