How do you prove this statement in geometry?

  1. 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. jcsd
  3. Office_Shredder

    Office_Shredder 4,487
    Staff Emeritus
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    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
  4. 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.
  5. Sorry, I meant positive measure.
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