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Consider a simplicial complex $C$. On its support $$|C|=\lbrace \alpha = \sum_{v\in C}\alpha_{v}v \mid 0\leq \alpha_{v} \leq 1 , \sum_{v\in C}\alpha_{v} =1\mbox{ and }v|{\alpha_{v}} \neq 0\mbox{ is a simplex in }C \rbrace $$ there is a metric, the barycentric metric, defined by $d(\alpha,\beta)=\sqrt{\sum_{v\in C}(\alpha_{v} - \beta_{v})^{2}}$. Is this metric complete ?

Greetings Ben

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closed as off topic by Misha, Benoît Kloeckner, Lee Mosher, Oscar Randal-Williams, Chris Gerig Jun 11 '13 at 16:21

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This doesn't look like research level… Please try another forum (e.g. MathStackExchange). –  Loïc Teyssier Jun 10 '13 at 7:48
    
Benjamin - some extra context might be helpful. Do you want to place any restrictions on $C$? Is it finite-dimensional, for instance? –  HJRW Jun 10 '13 at 12:21

1 Answer 1

No, this metric need not be complete. Let $C$ denote the complex with vertex set $\mathbb{N}$, where the simplices are the nonempty finite subsets. Thus, the points of $|C|$ are sequences of nonnegative real numbers that are eventually zero and sum to one. Consider the sequences \begin{align*} a_0 &= (1/2,1/2,0,0,0,0,\dotsc) \\\\ a_1 &= (1/2,1/4,1/4,0,0,0,\dotsc) \\\\ a_2 &= (1/2,1/4,1/8,1/8,0,0,\dotsc) \\\\ a_3 &= (1/2,1/4,1/8,1/16,1/16,0,\dotsc) \end{align*} and so on. Then $(a_n)_{n=0}^\infty$ is a Cauchy sequence in $|C|$ that does not converge.

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