Plane Separation

IF set of points P1 and P2 are convex subsets of a metric geometry, prove that P1 union P2 is convex.

plane Separation

If A,B,C are noncollinear in a metric geometry prove that triangle ABC is convex. I need to understand what this means and how to do a proof.

Geometry

Prove or disprove: If d0 and d1 are diestance functions on S, s is > or = to 0 and t > 0 then sd0 +td1 is also a distance function on S. I do not even know what this question means! Please help.

taxi cab

Prove or disprove that the Taxicab distance dt satisfies the triangle inequality.

Geometry -- Rectangle

Q: If you have a string of Christmas lights that is 100 feet long, and, you want to hang it in the form of a rectangle that its length is twice its width or more; what is the range of the width? [Hint: This means that we need the lowest and highest values that the width can be.]