Open problems in geometry
We present some open problems in convex geometry.
Examples. Let us consider an arbitrary convex closed curve, and the center of mass of the body, which corresponds to the domain inside the given curve. The lines going through this center of mass intersect the given curve twice, and they are called diameters. The smallest diameter is represented by d, and the maximum diameter represented by D. (i) If L is the length of the given curve, then L / D ≤ π ≤ L / d; (ii) Moreover, the first inequality becomes equality if and only if the second inequality becomes equality if and only if the given curve is a circle; (iii) If the area of the domain inside the given curve is A, then (iv) L d ≤ 4 A ≤ L D.
Several open problems in convex geometry appeared in some publications.