Separability of set-valued data sets and existence of support hyperplanes in the support function machine

The support function machine (SFM) has been shown to be effective in separating set-valued data sets. However, in SFM, the separability of set-valued data and the existence of support hyperplanes, which can provide useful guidance for improving algorithms for use in applications, have not been discussed in theory. Therefore, in this paper, we firstly discuss the problem of whether the linearly separable set-valued data in R^d are still linearly separable after being mapped into the infinite-dimensional Banach space C(S) by support functions. Secondly, we discuss the problem of whether the linearly inseparable set-valued data in R^d are linearly separable after being mapped into C(S). If not, in which situations are they linearly separable? Thirdly, we discuss the existence of support hyperplanes in SFM. Finally, two experiments with set-valued data sets are provided to verify the reasoning in the above discussions and the correctness of their conclusions.