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We establish new sharp inequalities of Poincar\'{e} or log-Sobolev type, ongeodesically-convex weighted Riemannian manifolds $(M,\mathfrak{g},\mu)$ whose(generalized) Ricci curvature $Ric_{\mathfrak{g},\mu,N}$ with effectivedimension parameter $N\in (-\infty,\infty]$ is bounded from below by a constant$K\in\mathbb{R}$, and whose diameter is bounded above by $D\in (0,\infty]$(Curvature-Dimension-Diameter conditions $CDD(K,N,D)$). To this end weestablish a general method which complements the `localization' theorem whichhas recently been established by B. Klartag. Klartag's Theorem is based onoptimal transport techniques, leading to a disintegration of the manifoldmeasure into marginal measures supported on geodesics of the manifold. Thisleads to a reduction of the problem of proving a n-dimensional inequality intoan optimization problem over a class of measures with 1-dimensional supports.In this work we firstly develop a general approach which leads to a reductionof this optimization problem into a simpler optimization problem, on a subclassof `model measures'. This reduction is based on functional analytic techniques,in particular a classification of extreme points of a specific subset ofmeasures, and showing that the solution to the optimization problem is attainedon this set of extreme points. Finally we solve the optimization problemsassociated with the Poincar\'{e}, p-Poincar\'{e} and the log-Sobolevinequalities subject to specific $CDD(K,N,D)$ conditions. Notably, we prove newsharp Poincar\'{e} inequalities for $N\in (-\infty,0]$. We find that for $N\in(-1,0]$ the characterization of the sharp lower bound on the Poincar\'{e}constant is of different nature; in addition we derive new lower bounds on thelog-Sobolev constant under $CDD(K,\infty,D)$ conditions where $K\in\mathbb{R}$and $D\in (0,\infty]$, which up to numeric constants are best possible.

Functional Inequalities on Weighted Riemannian Manifolds Subject to Curvature-Dimension Conditions