We study hidden geometry of Bier spheres Bier(K) = K * ? K? by describing their natural geometric realizations, compute volume, describe an effective criterion for polytopality, and associate to a coarsening Fan(K) the Braid fan. also establish connection maximal volume with recent generalizations classical Van Kampen-Flores theorem clarify role in theory generalized permutohedra.

We prove some results on maximal elements using the KKM-map principle.

In this paper, we discuss two variants of the generalized nonlinear vector variational-like inequality problem. We provide their solutions by adopting topological approach. Topological properties such as compactness, closedness, and net theory are used in proof. The admissibility function space topology KKM-Theorem have played important role proving results.

In the twenties Brouwer established the well-known continuity theorem “every real function is locally uniformly continuous”, [Brouwer 1924, Brouwer 1924a, Brouwer 1927]. From this theorem one immediately concludes that the continuum is indecomposable (unzerlegbar), i.e. if R = A∪B and A∩B = ∅ (denoted by R = A+B), then R = A or R = B. Brouwer deduced the indecomposability directly from the fan ...