Fuzzy Clustering Methods for Compact Sets via Star-Shaped Formalization

• A robust fuzzy k-means method is developed for clustering star-shaped sets. • The approach is extended to compact sets integrating mixed geometric information. • A tailored dissimilarity combines radial functions with numerical descriptors. • A controlled toy example shows the benefits of mixed and robust formulations. • The methods effectively cluster osteosarcoma cells and isolate atypical shapes. Fuzzy clustering methods for compact sets are proposed by combining their star-shaped representations with numerical geometric descriptors. The classical fuzzy k-means algorithm, previously extended to the star-shaped setting through a distance that accounts for both the centers and their radial functions, is further enhanced here with a robust extension that incorporates a noise cluster to reduce the impact of atypical sets. The same ideas are then transferred to compact sets carrying mixed-type information by means of a dissimilarity measure that blends functional and real-valued components, giving rise to basic and robust mixed-data variants. The approach is illustrated with osteosarcoma cell morphology, where integrating shape-based and numerical features offers a more informative description of cellular variability and supports the identification of unusual cells.