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Information geometry is a branch of mathematics that applies the techniques of differential geometry to the field of probability theory. This is done by taking probability distributions for a statistical model as the points of a Riemannian manifold, forming a statistical manifold. The Fisher information metric provides the Riemannian metric.

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About optimization with Renyi divergence

For your first question, the MSc thesis by Manuel Gil provides/derives closed form expressions for Renyi divergences associated with several of the continuous distributions.
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