# How do you prove this statement in geometry?

1. May 13, 2013

### phoenixthoth

A polygon with nonnegative area cannot be formed with fewer than 3 points.
A polyhedra with nonnegative volume cannot be formed with fewer than 4 points.
A hyperspace with nonnegative measure cannot be formed with fewer than n points.

What I mean by "3 points" is that the cardinality of the set of vertices is 3.

2. May 13, 2013

### Office_Shredder

Staff Emeritus
Actually a single point is a polygon of zero area which is nonnegative.

When you say "prove this statement in geometry" are you saying "prove this statement, which has to do with geometry" (because a proof using linear algebra is pretty easy) or are you asking "prove this statement using geometry" which I think would be significantly harder

3. May 20, 2013

### level1807

Take any n-1 points and build a small prism based on the polygon they create in a n-1-dimensional hyperplane (the polygon has to be completely inside the prism). Prism's volume is not more that S*h, where S is the "area"(n-1-dimensional) of the polygon and h is the length of the edge not coplanar with the hyperplane. Now it's clear that S*h tends to 0 as h tends to 0. By definition it means that the set of points of the polygon has "zero Lebesgue measure".
I hope you mentioned the word "cardinality" for no reason, because in an infinite-dimensional space volume as we're used to it (Lebesgue measure) doesn't exist.

4. May 8, 2015

### phoenixthoth

Sorry, I meant positive measure.