MathDB

Consider the closed plane curves

C_i

and

C_o,

their respective lengths

|C_i|

and

|C_o|,

the closed surfaces

S_i

and

S_o,

and their respective areas

|S_i|

and

|S_o|.

Assume that

C_i

lies inside

C_o

and

S_i

inside

S_o.

(Subscript

i

stands for "inner,"

o

for "outer.") Prove the correct assertions among the following four, and disprove the others.
(i) If

C_i

is convex,

|C_i| \le |C_o|.

(ii) If

S_i

is convex,

|S_i| \le |S_o|.

(iii) If

C_o

is the smallest convex curve containing

C_i,

then

|C_o| \le |C_i|.

(iv) If

S_o

is the smallest convex surface containing

S_i,

then

|S_o| \le |S_i|.

You may assume that

C_i

and

C_o

are polygons and

S_i

and

S_o

polyhedra.

Problem Statement