Abstract
To any compact set definable in an o-minimal structure, we associate a signed mea- sure, called scalar curvature measure. This generalizes the concept of scalar curvature on Rie- mannian manifolds. The main result states that, if the definable set is an Alexandrov space with curvature bounded below by k, then the scalar curvature measure is bounded below by kmðm � 1Þ volmð�Þ , where m is the dimension of the space and volmð�Þ the m-dimensional volume. This is a non-trivial generalization of a fact from dierential geometry. The proof combines techniques from o-minimal theory and from Alexandrov space theory. The back- ground of the definition of scalar curvature measure is given in the second part of the paper, where it is related to integral geometry and expressed by geometric quantities.
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