Simpliciality of strongly convex problems

A multiobjective optimization problem is C-r simplicial if the Pareto set and the Pareto front are C-r diffeomorphic to a simplex and, under the C-r diffeomorphisms, each face of the simplex corresponds to the Pareto set and the Pareto front of a subproblem, where 0 < r < infinity. In the paper titled "Topology of Pareto sets of strongly convex problems", it has been shown that a strongly convex C-r problem is Cr-1 simplicial under a mild assumption on the ranks of the differentials of the mapping for 2 < r < infinity. On the other hand, in this paper, we show that a strongly convex C-1 problem is C-0 simplicial under the same assumption. Moreover, we establish a specialized transversality theorem on generic linear perturbations of a strongly convex C-r mapping (r >= 2). By the transversality theorem, we also give an application of singularity theory to a strongly convex C-r problem for 2 < r < infinity.