Weak sharp minima for piecewise linear multiobjective optimization in normed spaces

In a general normed space, we consider a piecewise linear multiobjective optimization problem. We prove that a cone-convex piecewise linear multiobjective optimization problem always has a global weak sharp minimum property. By a counter example, we show that the weak sharp minimum property does not necessarily hold if the cone-convexity assumption is dropped. Moreover, under the assumption that the ordering cone is polyhedral, we prove that a (not necessarily cone-convex) piecewise linear multiobjective optimization problem always has a bounded weak sharp minimum property.