Largest Small n-Polygons: Numerical Results and Conjectured Optima

LSP(n), the largest small polygon with n vertices, is defined as the polygon of unit diameter that has maximal area A(n). Finding the configuration LSP(n) and the corresponding A(n) for even values n >= 6 is a long-standing challenge that leads to an interesting class of nonlinear optimization problems. We present numerical solution estimates for all even values 6 <= n <= 80, using the AMPL model development environment with the LGO nonlinear solver engine option. Our results compare favorably to the results obtained by other researchers who solved the problem using exact approaches (for 6 <= n <= 16), or general purpose numerical optimization software (for selected values from the range 6 <= n <= 100) using various local nonlinear solvers. Based on the results obtained, we also provide a regression model based estimate of the optimal area sequence {A(n)} for n >= 6.

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Research Report, Department of Industrial and Systems Engineering, Lehigh University, Bethlehem PA, USA. Submitted for publication.

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