Convex Functions and Their Applications: A Contemporary Approach

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A fixed-point approach to barycenters in Wasserstein space (2015, background): “By Alexandrov’s Theorem (see, e.g., Theorem 3.11.2 in [24] or Theorem 1 in Section 6.4 in [18]) we have that for µ -almost every x there is a symmetric, positive semideﬁnite matrix, which we denote ∇ 2 ϕ j ( x ) such that for every ε > 0 there exists δ > 0 such that (cid:107) y − x (cid:107) ≤ δ…”
A new representation for multivariate tail probabilities (2013, neutral): “κ(δ 1 +δ 2,ε 1+ε 2) ≤κ(δ 1,ε 1)+κ(δ 2,ε 2), that is, κ is subadditive. Homogeneous order 1 functions deﬁned on convex cones are convex if and only if they are subadditive (e.g., Niculescu and Persson [16], Lemma 3.6.1). In terms of λ, we have (δ 1+δ 2+ε 1+ε 2)λ δ 1+δ 2 δ 1+δ 2+ε 1+ε 2 ≤(δ 1+ε 1)λ ”