In the first part of this work, a convex partition of a compact subset is constructed. Minimum-length surrounding curve and minimum-area surrounding surfaces for a compact set are constructed too. In the second part, one writes the perimeter of an ellipse as the sum of an alternate series. On the other hand, we deduce related “sandwich” inequalities for the perimeter, involving Jensen’s inequality and logarithmic function respectively. We discuss the values of the ordinate of the gravity center of the upper semiellipse at the ends of the positive semiaxes, in terms of the scale ratio .
Almkvist, G. and Berndt, B. (1988) Gauss, Landen, Ramanujan, the Arithmetic Geometric Mean, Ellipses, π, and the Ladies Diary. The American Mathematical Monthly, 95, 585-608. http://dx.doi.org/10.2307/2323302
Barnard, R.W., Pearce, K. and Schovanec, L. (2001) Inequalities for the Perimeter of an Ellipse. Journal of Mathemat- ical Analysis and Applications, 260, 295-306. http://dx.doi.org/10.1006/jmaa.2000.7128
Deville, R., Fonf, D. and Hájek, P. (1998) Analytic and Polyhedral Approximation of Convex Bodies in Separable Polyhedral Banach Spaces. Israel Journal of Mathematics, 105, 139-154. http://dx.doi.org/10.1007/BF02780326