Focus Area 3: Topological connectivity and diameter of Discrete Structures A k-fan in the plane is a point x∈ℝ2 and k halflines starting from x. There are k angular sectors σ 1,…,σ k between consecutive halflines. The k-fan is convex if every sector is convex. A (nice) probability measure μ is equipartitioned by the k-fan if μ(σ i )=1/k for every sector. One of our results: Given a nice probability measure μ and a continuous function f defined on sectors, there is a convex 5-fan equipartitioning μ with f(σ 1)=f(σ 2)=f(σ 3).