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Martin Sleziak
Martin Sleziak
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You can isometrically embed any metric space into a Banach space via the Arens-Eells theorem (original: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1103043959https://projecteuclid.org/euclid.pjm/1103043959 simpler proof by E. Michael: http://www.jstor.org/stable/2034516?origin=JSTOR-pdf ). This embedding is, in some sense, canonical. Convex consecones are well-defined in Banach spaces, so you could say that a point x is in the convex cone generated by x_1, ... x_n in the original metric space if f(x) is in the cone of f(x_1), ..., f(x_n) -- where f is the Arens-Eells embedding.

You can isometrically embed any metric space into a Banach space via the Arens-Eells theorem (original: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1103043959 simpler proof by E. Michael: http://www.jstor.org/stable/2034516?origin=JSTOR-pdf ). This embedding is, in some sense, canonical. Convex conse are well-defined in Banach spaces, so you could say that a point x is in the convex cone generated by x_1, ... x_n in the original metric space if f(x) is in the cone of f(x_1), ..., f(x_n) -- where f is the Arens-Eells embedding.

You can isometrically embed any metric space into a Banach space via the Arens-Eells theorem (original: https://projecteuclid.org/euclid.pjm/1103043959 simpler proof by E. Michael: http://www.jstor.org/stable/2034516?origin=JSTOR-pdf ). This embedding is, in some sense, canonical. Convex cones are well-defined in Banach spaces, so you could say that a point x is in the convex cone generated by x_1, ... x_n in the original metric space if f(x) is in the cone of f(x_1), ..., f(x_n) -- where f is the Arens-Eells embedding.

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Aryeh Kontorovich
Aryeh Kontorovich
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You can isometrically embed any metric space into a Banach space via the Arens-Eells theorem (original: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1103043959 simpler proof by E. Michael: http://www.jstor.org/stable/2034516?origin=JSTOR-pdf ). This embedding is, in some sense, canonical. Convex conse are well-defined in Banach spaces, so you could say that a point x is in the convex cone generated by x_1, ... x_n in the original metric space if f(x) is in the cone of f(x_1), ..., f(x_n) -- where f is the Arens-Eells embedding.