Date of Graduation
Master of Science in Mathematics
k-simplex, n-cube, Cayley-Menger determinants, Bernstein polynomials, Monte Carlo methods
Algebraic Geometry | Discrete Mathematics and Combinatorics | Geometry and Topology
Within an n-dimensional unit cube, a number of k-dimensional simplices can be formed whose vertices are the vertices of the n-cube. In this thesis, we analyze the average measure of a k-simplex in the n-cube. We develop exact equations for the average measure when k = 1, 2, and 3. Then we generate data for these cases and conjecture that their averages appear to approach nk/2 times some constant. Using the convergence of Bernstein polynomials and a k-simplex Bernstein generalization, we prove the conjecture is true for the 1-simplex and 2-simplex cases. We then develop a generalized formula for the average measure of the k-simplex in the n-cube and prove the average is asymptotic to nk/2 √ k+1/2kk!.
© John A. Carter
Carter, John A., "The Average Measure of a k-Dimensional Simplex in an n-Cube" (2018). MSU Graduate Theses. 3247.