Bisections of Mass Assignments Using Flags of Affine Spaces

We use recent extensions of the Borsuk–Ulam theorem for Stiefel manifolds to generalize ham sandwich mass assignments. A k-dimensional assignment continuously imposes a measure on each affine subspace $${\mathbb {R}}^d$$ . Given finite collection assignments different dimensions, one may ask if there is some sequence subspaces $$S_{k-1} \subset S_k \ldots S_{d-1} {\mathbb such that $$S_i$$ bisects all $$S_{i+1}$$ every i. show it possible do so whenever number dimensions $$(k,\ldots ,d)$$ permutation extend previous work and central transversal theorem. also study problem halving several families $$(d-k)$$ -dimensional spaces using $$(k-1)$$ contained in translate fixed space. For $$k=d-1$$ , results can be interpreted as dynamic theorems moving points.