#### Date of Graduation

Spring 2018

#### Degree

Master of Science in Mathematics

#### Department

Mathematics

#### Committee Chair

Les Reid

#### Keywords

k-simplex, n-cube, Cayley-Menger determinants, Bernstein polynomials, Monte Carlo methods

#### Subject Categories

Algebraic Geometry | Discrete Mathematics and Combinatorics | Geometry and Topology

#### Abstract

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 n^{k/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 n^{k/2} √ k+1/2^{k}k!.

#### Copyright

© John A. Carter

#### Recommended Citation

Carter, John A., "The Average Measure of a k-Dimensional Simplex in an n-Cube" (2018). *MSU Graduate Theses*. 3247.

https://bearworks.missouristate.edu/theses/3247

**Open Access**

#### Included in

Algebraic Geometry Commons, Discrete Mathematics and Combinatorics Commons, Geometry and Topology Commons