AN ANTIPODAL THEOREM FOR PARAMETRIC OPTIMIZATION PROBLEMS

Kawasaki (2023) introduced Borsuk- Ulam's theorem to nonlinear optimization. It applied Borsuk-Ulam's theorem to an n-tuple of parametric optimization problems with parameter u ∈ S^n, and presented an antipodal theorem for them. Further, it showed that given n convex sets in ℝ^n, it is possible to divide each width in half with a hyperplane. In this paper, we take another approach, which weakens the assumptions, simplifies the proofs, and includes a result beyond the scope of Kawasaki (2023). Further, we add a new insight to the ham-sandwich theorem.