## Optimal discrete measures for Riesz potentials

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- by S. V. Borodachov, D. P. Hardin, A. Reznikov and E. B. Saff PDF
- Trans. Amer. Math. Soc.
**370**(2018), 6973-6993 Request permission

## Abstract:

For weighted Riesz potentials of the form $K(x,y)=w(x,y)/$ $|x-y|^s$, we investigate $N$-point configurations $x_1,x_2, \ldots , x_N$ on a $d$-dimensional compact subset $A$ of $\mathbb {R}^p$ for which the minimum of $\sum _{j=1}^NK(x,x_j)$ on $A$ is maximal. Such quantities are called $N$-point Riesz $s$-polarization (or Chebyshev) constants. For $s\geqslant d$, we obtain the dominant term as $N\to \infty$ of such constants for a class of $d$-rectifiable subsets of $\mathbb {R}^p$. This class includes compact subsets of $d$-dimensional $C^1$ manifolds whose boundary relative to the manifold has $d$-dimensional Hausdorff measure zero, as well as finite unions of such sets when their pairwise intersections have measure zero. We also explicitly determine the weak-star limit distribution of asymptotically optimal $N$-point configurations for weighted $s$-polarization as $N\to \infty$.## References

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## Additional Information

**S. V. Borodachov**- Affiliation: Department of Mathematics, Towson University, Towson, Maryland 21252
- MR Author ID: 656604
- Email: sborodachov@towson.edu
**D. P. Hardin**- Affiliation: Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37235
- MR Author ID: 81245
- ORCID: 0000-0003-0867-2146
- Email: doug.hardin@vanderbilt.edu
**A. Reznikov**- Affiliation: Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37235
- Address at time of publication: Department of Mathematics, Florida State University, Tallahassee, Florida 32306
- MR Author ID: 895080
- Email: reznikov@math.fsu.edu
**E. B. Saff**- Affiliation: Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37235
- MR Author ID: 152845
- Email: edward.b.saff@vanderbilt.edu
- Received by editor(s): October 13, 2016
- Received by editor(s) in revised form: February 6, 2017
- Published electronically: April 17, 2018
- Additional Notes: This research was supported, in part, by the U.S. National Science Foundation under the grant DMS-1412428 and DMS-1516400
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 6973-6993 - MSC (2010): Primary 31C20, 31C45; Secondary 28A78
- DOI: https://doi.org/10.1090/tran/7224
- MathSciNet review: 3841839