Re: D1 polytope

On Dec 25, 2:29 pm, beewo...@xxxxxxxxxxx wrote:
On Dec 25, 6:58 am, Denis Feldmann <denis.feldmann.sanss...@xxxxxxx>
wrote:



beewo...@xxxxxxxxxxx a écrit :

On Dec 25, 6:25 am, Denis Feldmann <denis.feldmann.sanss...@xxxxxxx>
wrote:
beewo...@xxxxxxxxxxx a écrit :

On Dec 23, 1:30 pm, Axel Harvey <x...@xxxxxxxxxxxx> wrote:
Is a line segment considered a 1-dimensional polytope?
Axel,
A line segment is not a polytope in one dimension.  The end points of
a line segment is.  Just as a triangle consists of its sides and not
its interior, a 1-D polytope consists of the end points of a line
segment, but not the points between those segments.
Perhaps you should check before answering (or read the whole thread)....

- MO- Hide quoted text -
- Show quoted text -

Perhaps.  Mariano asked a question whose answer would indicate that a
1-D polytope is just two points.  But then Axel responded by saying he
was not a mathematician.  Which indicated to me that he was looking
for an intuitive answer to his question, rather than a formal one.
David chimed in nicely by writing (allow me to paraphrase) that if
Axel was using the term "polytope" he should know the definition of
it.  Well, that did not seem to me to be of much help, because it
seemed that Axel was seeking to check on his understanding of the
definition of polytope. Spudnik's third reply seem to offer some
correct and more or less relavent information, but again, like
Mariano's, it was steeped in formal mathematical jargon rather than
offering an intuitive answer.  In reply to Mariano's third reply, Axel
reitierated that he is not a mathematician, and indicated that he was
looking for an intuitive answer.

So I offered an intuitive answer that was the "Polytopes for Dummies"
version of Mariano's.

- MO

Except your answer is wrong (or at least doesn't agree with the usual
definition, which is anyway easier) : polytopes "should" be convex hulls
, not skeletons, or more generally objects of the same dimension that
the embedding space, not one lower. Otoh, it is hard to define, say,
concave polyhedrons that way, and next, we enter the simplicial complex
definitons, which are surely not appropriate for a beginner :-)- Hide quoted text -

- Show quoted text -

My understanding is that a triangle is three connected line segments
and does not include its interior.  You are telling me that a three-
sided polygon is a triangle ~and~ its interior, rather than just the
triangle. Is that right?

"Triangle" can mean several things, depending on the context.

Dennis is indeed correct that your answer to Axel was wrong.
A segment *is* a polytope. And, in the context of polytopes,
a triangle most often means the "filled" triangle, and is
pretty much the same thing as a 2-simplex---indeed, one speaks
of "triangulating" a surface, for example.

-- m
.

Relevant Pages

  • Re: D1 polytope
    ... A line segment is not a polytope in one dimension. ...  But then Axel responded by saying he ... My understanding is that a triangle is three connected line segments ...
    (sci.math)
  • Re: D1 polytope
    ... A line segment is not a polytope in one dimension. ... But then Axel responded by saying he ... Except your answer is wrong: polytopes "should" be convex hulls, not skeletons, or more generally objects of the same dimension that the embedding space, not one lower. ...
    (sci.math)
  • Re: D1 polytope
    ...  Just as a triangle consists of its sides and not ... segment, but not the points between those segments. ... 1-D polytope is just two points. ... But then Axel responded by saying he ...
    (sci.math)
  • Re: How to get the intersection point coordinate between polygon and segment
    ... for example(a is a triangle a(i,:) is the ith coordinate of the triangle.b is an segment,bis the side point coordinate): ... %find intersection points of line segment with polytope ...
    (comp.soft-sys.matlab)
  • Re: D1 polytope
    ... One of the standard definitions of "polytope" is ... "closed convex hull of a finite set of points in R^n". ... a segment is a polytope or not: ... your not knowing what the definition of a polytope is ...
    (sci.math)