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4. Discreteness and dimension reduction of minimal measures

    1.     Consider the problem of minimizing $$ I(\mu) = \iint_\Omega K(x,y) \, d\mu(x)d\mu(y) $$ over probability measures on a fixed compact (often convex) set.

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      Problem 4.1.

      [Dima Bilyk] Under which assumptions the measures minimizing $I(\cdot)$ are discrete?


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          Problem 4.2.

          [Dima Bilyk] Prove that all minimizers of p-frame energy $$ I_p(\mu) = \iint_{\mathbb S^d} |x\cdot y|^p \, d\mu(x) d\mu(y) $$ over probability measures on the sphere $\mathbb S^d$ are discrete when $p>0$, $p\notin 2\mathbb N$.

              Cite this as: AimPL: Minimal energy problems with Riesz potentials, available at http://aimpl.org/energyrieszV.