S. Kondratyev, D. Vorotnikov. Spherical Hellinger-Kantorovich gradient flows, SIAM J. Math. Anal 51 (2019) 2053–2084.

On p. 2082 it is claimed that the entropy E₁ is geodesically convex in the conic Hellinger-Kantorovich space for positive alpha; however, this entropy is merely geodesically semiconvex.

S. Kondratyev, D. Vorotnikov.  Nonlinear Fokker-Planck equations with reaction as gradient flows of the free energy, J. Funct. Anal. 278 (2020) 108310.

In Theorem 2.9, the phrase "let U be a set of strictly positive functions having the property that ... converges to 0 in measure" can be replaced by "... converges to 0 in L^1". Indeed, if a sequence u_n for which the constant in (2.40) blows up converges in measure (and has bounded integral of E) but a concentration of mass happens (the integrals of u_n are bounded away from 0), then for any M there exists positive delta and natural N such that the integral of u_n w.r.t. to the set O_n where u_n is larger than M is larger than delta for every n bigger than N. If M is sufficiently large, the corresponding fitnesses f_n are bounded away from 0 on O_n. Hence for the sequence in question the constant in (2.40) cannot blowup. Indeed, the LHS is bounded by assumption; thus it is  controlled by the integral of u f^2 w.r.t. O_n alone, which is smaller than the RHS. This contradiction shows that w.l.o.g. there are no concentrations. 

This observation makes the proof of Theorem 1.8 smoother and fixes some gaps.