The passage discusses the proof of an inequality for each term in a summation, assuming the thesis is wrong. By squaring and simplifying the expression, it is shown that the inequality leads to a contradiction, indicating that equality can only occur if the corresponding sides of the polygons are parallel. The text also proves that the minimal polygon must have at least one vertex on the boundary of the domain, and that all local minima are global due to the convexity properties of the domain and the function. Additionally, it is shown that any convex linear combination of minima is also a minimum, and Jensen's relation holds with equality if and only if the sides of the polygons are parallel. The minimal reduced polygon is unique, as any nonactive constraints do not affect the position of the vertices between noncollinear sides.The passage discusses the proof of an inequality for each term in a summation, assuming the thesis is wrong. By squaring and simplifying the expression, it is shown that the inequality leads to a contradiction, indicating that equality can only occur if the corresponding sides of the polygons are parallel. The text also proves that the minimal polygon must have at least one vertex on the boundary of the domain, and that all local minima are global due to the convexity properties of the domain and the function. Additionally, it is shown that any convex linear combination of minima is also a minimum, and Jensen's relation holds with equality if and only if the sides of the polygons are parallel. The minimal reduced polygon is unique, as any nonactive constraints do not affect the position of the vertices between noncollinear sides.