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Tverberg’s theorem states that any set of (q-1)(d+1)+1 points in d-dimensional Euclidean space can be partitioned into q subsets whose convex hulls intersect. This is topologically equivalent to saying any continuous map from a (q-1)(d+1)-dimensional simplex to d-dimensional Euclidean space has q disjoint faces whose images intersect, given that q is a prime power. These continuous functions have a Fourier decomposition, which admits a Tverberg partition when all of the Fourier coefficients, except the constant coefficient, are zero. We have been working with continuous functions where all of the Fourier coefficients except the constant and one other coefficient are zero. This results in there being q disjoint faces whose images, rather than intersecting, contain points that form a regular q-gon. The van Kampen-Flores theorem increases the dimension of the simplex of the topological Tverberg theorem to restrict the dimension of the disjoint faces. Similarly, we found that if 2d(q-2)+1
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Leiner, Leah, "Polygonal Analogues to the Topological Tverberg and van Kampen-Flores Theorems" (2019). Senior Projects Spring 2019. 204.
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