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Properties of a volume in 3D Euclidean space

  1. Nov 10, 2009 #1

    I am writing a small report and trying to be mathematically accurate in my terminology- I am trying to describe an arbitrary volume of gas, but this volume must (1) not have any holes (or bubbles where the gas cannot go) in it, and (2) must be one single volume, so a gas molecule from any point can reach any other point without jumping.

    Am I right in thinking the second condition is satisfied if the volume is "connected"?
    What is the first condition described by?

    It can be any shape, just no holes.


    edit- Also I am considering a third condition, that (3) any straight line passing through the volume only crosses two boundaries- for example a skewer through a donut can cross four times if you poke it through a line in the 'plane' of the donut. Is there a word for this? I think it may relate to the curvature of the edge surface, must be positive or zero everywhere?
  2. jcsd
  3. Nov 10, 2009 #2
    Genus 0.
    You don't want that to be zero if you only want two points. 'Strictly convex' is the term you are looking for.
  4. Nov 10, 2009 #3
    How about a box, though, the curvature should be zero everywhere, and that's a realistic example for me, where you can't have more than two points?
  5. Nov 10, 2009 #4
    Then use 'convex'. This includes zero curvature. And doesn't need 'genus 0'.
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