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Optimization with convex constraints and convex objectives; notions related to convex optimization such as sub-gradients, normal cones, separating hyperplanes

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Convergence of the infima of convex functions on $\mathbb{R}^m$

Any thoughts on proving the following statement, which is a generalization of the result in convergence of the infima of convex functions from domain $\mathbb{R}$ to $\mathbb{R}^m$ and also Theorem 1 …
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