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From Wikipedia:

an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices.

I was wondering if the definition is equivalent to say a simplex is synonym of a convex polytope?

Is simplex defined only for $\mathbb{R}^n$, not for other more general spaces?

Thanks and regards!

J W
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Tim
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4 Answers4

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A simplex (in $\mathbb{R}^{n}$) is a special case of a convex polytope, namely one with exactly $n+1$ extremal points (the convex hull of $n+1$ points in general position - that is to say they don't lie in a $(n-1)$-dimensional affine subspace). An example of a convex polytope which isn't a simplex is given by any one of the platonic solids except the tetrahedron.

Yes, there are more general notions of simplices. In a geodesic metric space, for example, it makes sense to speak of convex hulls. In hyperbolic $n$-space or on the $n$-dimensional sphere one can then define a hyperbolic or spherical simplex as the convex hull $(n+1)$ points that don't lie in a totally geodesic submanifold (and that are not too far apart from each other in the case of a sphere).

t.b.
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  • Thanks! I remember that convex hull makes sense for a subset of a vector space with ordered base field. Isn't it? Is a geodesic metric space such vector space? If not, how does it make sense to speak of convex hulls there? – Tim May 01 '11 at 17:15
  • @Tim: In connection with vector spaces I've only seen convex hulls for real or complex vector spaces. A geodesic metric space is a metric space $(X,d)$ with the following property: For any two points $x,y$ there exists an isometry $\gamma: [0,d(x,y)] \to X$ such that $\gamma(0) = x$ and $\gamma(d(x,y)) = y$ (such an isometry is interpreted as a geodesic parameterized by arc length). A convex set $C$ is then a set such that for all $x,y \in C$ we have $\gamma([0,d(x,y)]) \subset C$ for all geodesics $\gamma$. The convex hull of $S$ is then the intersection of all convex sets containing $S$. – t.b. May 01 '11 at 17:22
  • Thanks! So are these two unrelated definitions of convex hull? – Tim May 01 '11 at 17:27
  • @Tim: No. For $\mathbb{R}^n$ they are equivalent. – t.b. May 01 '11 at 17:28
  • @t.b. Do you have a reference on general simplexes? – hrkrshnn Jun 02 '16 at 15:04
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The definition is a little unclear. In order for the convex hull of $n+1$ vertices in $\mathbb{R}^n$ to actually be $n$-dimensional, the vertices must be in general position: that is, they should not lie in a proper affine subspace of $\mathbb{R}^n$. For example, a square is a convex polytope which is the convex hull of $4$ vertices, but it is not a $3$-simplex: a $3$-simplex is a tetrahedron.

I suppose the answer to the last question is "no": when people talk about geometric simplices they are definitely referring to the ones in $\mathbb{R}^n$, although some people talk about abstract simplices in the sense of abstract simplicial complexes.

Qiaochu Yuan
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  • Thanks! (1) So can I say a simplex is defined as a convex polytope whose vertices do not liey in a proper affine subspace of $\mathbb{R}^n$? (2) is it defined for more general space than $\mathbb{R}^n$? – Tim May 01 '11 at 16:39
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    @Qiaochu: The definition isn't incomplete (though it could be a bit clearer); I think the fact that they are in general position is covered by the statement that the convex hull is in fact an $n$-dimensional polytope. – Zev Chonoles May 01 '11 at 16:39
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  • You need the condition that there are $n+1$ vertices, but yes. 2) A more general space than $\mathbb{R}^n$ will generally not have a notion of "convex hull."
    • – Qiaochu Yuan May 01 '11 at 16:40
  • @Zev: okay, that's fair. – Qiaochu Yuan May 01 '11 at 16:41
  • @Qiaochu: Here is a book that discusses the abstract notion of a convex hull. http://www.amazon.com/Theory-Structures-North-Holland-Mathematical-Library/dp/0444815058/ref=sr_1_1?s=books&ie=UTF8&qid=1304294054&sr=1-1 – Jay May 01 '11 at 23:57
  • @Qiaochu: Convex hull operators are everywhere. It may be that they are not recognized as such. In group theory we have the smallest group that contains a given set as well as the smallest coset that contains a given set. A similar statement can be made for rings. – Jay May 03 '11 at 15:58