The text presents a mathematical proof and algorithmic content. The proof demonstrates that an inequality derived from the difference of two polygon sides leads to a contradiction unless the sides are parallel, implying that the minimal polygon must have vertices on the boundary of the domain. It also shows that the minimal polygon's vertices between noncollinear sides must satisfy a bisection property. The proof further establishes that for convex domains and functions, all local minima are global, and any convex combination of minima is also a minimum. It concludes that the minimal reduced polygon is unique. The text also includes an algorithm for numerically inverting Laplace transforms, known as Algorithm 368. This algorithm uses a method of extrapolation to the limit to approximate the inverse Laplace transform. It is based on the expectation of the function with respect to a probability density function and involves a linear combination of approximations. The algorithm's accuracy depends on the number of significant figures and the value of N, which is the number of P-values used. The algorithm is tested with 50 known transforms and is found to be accurate for smooth functions. Additionally, the text includes algorithms for generating random numbers satisfying the Poisson distribution and a general random number generator. The Poisson distribution algorithm uses the rejection method and is efficient for small numbers of random numbers. The general random number generator uses transformation theory to generate random numbers from any probability density function, using 257 points and linear approximation for accuracy and speed. The text also includes a remark on Algorithm 282, which discusses the derivatives of functions like $e^x/x$, $\cos(x)/x$, and $\sin(x)/x$. The remark explains how to compute these derivatives accurately for large values of x and high-order derivatives. The text concludes with a remark on an efficient sorting algorithm with minimal storage, which was tested on various computers and found to be accurate. The algorithm uses dynamic bounds for arrays and is efficient for sorting integer arrays.The text presents a mathematical proof and algorithmic content. The proof demonstrates that an inequality derived from the difference of two polygon sides leads to a contradiction unless the sides are parallel, implying that the minimal polygon must have vertices on the boundary of the domain. It also shows that the minimal polygon's vertices between noncollinear sides must satisfy a bisection property. The proof further establishes that for convex domains and functions, all local minima are global, and any convex combination of minima is also a minimum. It concludes that the minimal reduced polygon is unique. The text also includes an algorithm for numerically inverting Laplace transforms, known as Algorithm 368. This algorithm uses a method of extrapolation to the limit to approximate the inverse Laplace transform. It is based on the expectation of the function with respect to a probability density function and involves a linear combination of approximations. The algorithm's accuracy depends on the number of significant figures and the value of N, which is the number of P-values used. The algorithm is tested with 50 known transforms and is found to be accurate for smooth functions. Additionally, the text includes algorithms for generating random numbers satisfying the Poisson distribution and a general random number generator. The Poisson distribution algorithm uses the rejection method and is efficient for small numbers of random numbers. The general random number generator uses transformation theory to generate random numbers from any probability density function, using 257 points and linear approximation for accuracy and speed. The text also includes a remark on Algorithm 282, which discusses the derivatives of functions like $e^x/x$, $\cos(x)/x$, and $\sin(x)/x$. The remark explains how to compute these derivatives accurately for large values of x and high-order derivatives. The text concludes with a remark on an efficient sorting algorithm with minimal storage, which was tested on various computers and found to be accurate. The algorithm uses dynamic bounds for arrays and is efficient for sorting integer arrays.