For essentially convex interaction kernels, a threshold condition on an auxiliary density guarantees existence, uniqueness, and compact support of minimizers in R^d and curved half-spaces; a counterexample shows the threshold fails for superharmonic kernels.
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Existence of minimizers for interaction energies with external potentials
For essentially convex interaction kernels, a threshold condition on an auxiliary density guarantees existence, uniqueness, and compact support of minimizers in R^d and curved half-spaces; a counterexample shows the threshold fails for superharmonic kernels.