An optimization problem on the sphere

We prove existence and uniqueness of the minimizer for the average geodesic distance to the points of a geodesically convex set on the sphere. This implies a corresponding existence and uniqueness result for an optimal algorithm for halfspace learning, when data and target functions are drawn from the uniform distribution.

💡 Research Summary

The paper studies the problem of minimizing the average geodesic distance from a point on the unit sphere Sⁿ⁻¹ to all points of a geodesically convex subset of the sphere, which is represented as the intersection of a closed proper convex cone K⊂ℝⁿ with the sphere. For a given cone K, the authors define the functional

 ψ_K(w) = ∫_{K∩Sⁿ⁻¹} g(ρ(w,y)) dσ(y),

where ρ(w,y) = arccos⟨w,y⟩ is the geodesic distance, σ is the normalized uniform measure on the sphere, and g: