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The authors prove that upper curvature bounds in the sense of Alexander can be improved locally. A new technical tool concerning an extension to metric spaces and semi-convex funtions of the property that compositions of harmonic maps with convex funtions

The authors prove that upper curvature bounds in the sense of Alexander can be improved locally. A new technical tool concerning an extension to metric spaces and semi-convex funtions of the property that compositions of harmonic maps with convex funtions are subharmonic is derived. In particular, the authors prove the following result: For c, C, κ, λ ∈ R, let X be a CAT(κ) space and let f : X → [c, C] be a Lipschitz continuous λ-convex funtion. Further, let Y = e X denote the conformally equivalent space. If κ − 2λ ≤ 0, then Y is CAT(κ) with κ = e−2C(κ − 2λ). If κ − 2λ > 0 and λ > 0, then Y is CAT(κ) with κ = e−2c(κ − 2λ).