It is shown that within the Lp-Brunn–Minkowski theory that Aleksandrov’s integral curvature has a natural Lp extension, for all real p. This raises the question of finding necessary and sufficient conditions on a given measure in order for it to be the Lp-integral curvature of a convex body. This problem is solved for positive p and is answered for negative p provided the given measure is even.

We show that the FI-homology of an FI-module can be computed via a Koszul complex. As an application, we prove that the Castelnuovo-Mumford regularity of a finitely generated torsion FI-module is equal to its degree.

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We study a partially linearized version of the relative trace formula for the arithmetic Gan--Gross--Prasad conjecture for the unitary group U(V). The linear factor in this relative trace formula provides an SL_2-symmetry which allows us to prove by induction the arithmetic fundamental lemma over Q_p when p is odd and p ≥ dim V.

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