论文标题
詹森（Jensen
Jensen's inequality in geodesic spaces with lower bounded curvature
论文作者
论文摘要
从Alexandrov的意义上讲，令$（m，d）$为一个可分离且完整的地理空间，曲率下限为$κ\ in \ Mathbb r $。令$μ$为$ m $上的borel概率度量，以便在\ Mathcal p_2（m）$中$μ\，并且至少有一个Barycenter $ x^{*} \ m $。我们表明，对于任何地理上的$α$ -CONVEX函数$ f：m \ to \ mathbb r $，对于$α\ in \ mathbb r $，不平等\ [f（x^*）\ le \ int_m（f - \fracα{2} d} d^2^2^2^2（x^*$;本地Lipschitz为$ x^*$，或$ l^1（μ）$。我们的证明依赖于Barycenters切线锥的特性以及在具有下限曲率的空间中半轴函数的梯度的存在。
Let $(M,d)$ be a separable and complete geodesic space with curvature lower bounded, by $κ\in \mathbb R$, in the sense of Alexandrov. Let $μ$ be a Borel probability measure on $M$, such that $μ\in\mathcal P_2(M)$, and that has at least one barycenter $x^{*}\in M$. We show that for any geodesically $α$-convex function $f:M\to \mathbb R$, for $α\in \mathbb R$, the inequality \[f(x^*)\le \int_M (f -\fracα{2}d^2(x^*,.))\,{\rm d}μ,\] holds provided $f$ is locally Lipschitz at $x^*$ and either positive or in $L^1(μ)$. Our proof relies on the properties of tangent cones at barycenters and on the existence of gradients for semi-concave functions in spaces with lower bounded curvature.
