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Added a clarification about implicit aspects of the question.
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Benoît Kloeckner
Benoît Kloeckner
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(As asked, I make my comment into an answer, completing it with the OP's comment)

As far as I know, this result was not known before Lott-Villani and Sturm's works; it is included as Corollary 0.14 in Lott-Villani's article.

As we do not know a lot of spaces satisfying the Ricci lower bounds defined by this theory, apart from manifolds and Alexandrov spaces, this is currently the most concrete application of these works.

Added: for clarity, I stress here the comment of Chris Gerig to Anton Petrunin's answer: Lott-Villani and Sturm's prove that the result holds even if the limiting manifold is of different dimension than the converging manifolds.

(As asked, I make my comment into an answer, completing it with the OP's comment)

As far as I know, this result was not known before Lott-Villani and Sturm's works; it is included as Corollary 0.14 in Lott-Villani's article.

As we do not know a lot of spaces satisfying the Ricci lower bounds defined by this theory, apart from manifolds and Alexandrov spaces, this is currently the most concrete application of these works.

(As asked, I make my comment into an answer, completing it with the OP's comment)

As far as I know, this result was not known before Lott-Villani and Sturm's works; it is included as Corollary 0.14 in Lott-Villani's article.

As we do not know a lot of spaces satisfying the Ricci lower bounds defined by this theory, apart from manifolds and Alexandrov spaces, this is currently the most concrete application of these works.

Added: for clarity, I stress here the comment of Chris Gerig to Anton Petrunin's answer: Lott-Villani and Sturm's prove that the result holds even if the limiting manifold is of different dimension than the converging manifolds.

Source Link
Benoît Kloeckner
Benoît Kloeckner
  • 14.4k
  • 1
  • 60
  • 106

(As asked, I make my comment into an answer, completing it with the OP's comment)

As far as I know, this result was not known before Lott-Villani and Sturm's works; it is included as Corollary 0.14 in Lott-Villani's article.

As we do not know a lot of spaces satisfying the Ricci lower bounds defined by this theory, apart from manifolds and Alexandrov spaces, this is currently the most concrete application of these works.