Problems(1)
Consider the closed plane curves Ci and Co, their respective lengths ∣Ci∣ and ∣Co∣, the closed surfaces Si and So, and their respective areas ∣Si∣ and ∣So∣. Assume that Ci lies inside Co and Si inside So. (Subscript i stands for "inner," o for "outer.") Prove the correct assertions among the following four, and disprove the others.(i) If Ci is convex, ∣Ci∣≤∣Co∣.
(ii) If Si is convex, ∣Si∣≤∣So∣.
(iii) If Co is the smallest convex curve containing Ci, then ∣Co∣≤∣Ci∣.
(iv) If So is the smallest convex surface containing Si, then ∣So∣≤∣Si∣.You may assume that Ci and Co are polygons and Si and So polyhedra. Putnam