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Maximizing Riesz capacity ratios: conjectures and theorems

Carrie Clark, Richard S. Laugesen

Abstract

A shape optimization program is developed for the ratio of Riesz capacities $\text{Cap}_q(K)/\text{Cap}_p(K)$, where $K$ ranges over compact sets in $\mathbb{R}^n$. In different regions of the $pq$-parameter plane, maximality is conjectured for the ball, the vertices of a regular simplex, or the endpoints of an interval. These cases are separated by a symmetry-breaking transition region where the shape of maximizers remains unclear. On the boundary of $pq$-parameter space one encounters existing theorems and conjectures, including: Watanabe's theorem minimizing Riesz capacity for given volume, the classical isodiametric theorem that maximizes volume for given diameter, Szegő's isodiametric theorem maximizing Newtonian capacity for given diameter, and the still-open isodiametric conjecture for Riesz capacity. The first quadrant of parameter space contains Pólya and Szegő's conjecture on maximizing Newtonian over logarithmic capacity for planar sets. The maximal shape for each of these scenarios is known or conjectured to be the ball. In the third quadrant, where both $p$ and $q$ are negative, the maximizers are quite different: when one of the parameters is $-\infty$ and the other is suitably negative, maximality is proved for the vertices of a regular simplex or endpoints of an interval. Much more is proved in dimensions $1$ and $2$, where for large regions of the third quadrant, maximizers are shown to consist of the vertices of intervals or equilateral triangles.

Maximizing Riesz capacity ratios: conjectures and theorems

Abstract

A shape optimization program is developed for the ratio of Riesz capacities , where ranges over compact sets in . In different regions of the -parameter plane, maximality is conjectured for the ball, the vertices of a regular simplex, or the endpoints of an interval. These cases are separated by a symmetry-breaking transition region where the shape of maximizers remains unclear. On the boundary of -parameter space one encounters existing theorems and conjectures, including: Watanabe's theorem minimizing Riesz capacity for given volume, the classical isodiametric theorem that maximizes volume for given diameter, Szegő's isodiametric theorem maximizing Newtonian capacity for given diameter, and the still-open isodiametric conjecture for Riesz capacity. The first quadrant of parameter space contains Pólya and Szegő's conjecture on maximizing Newtonian over logarithmic capacity for planar sets. The maximal shape for each of these scenarios is known or conjectured to be the ball. In the third quadrant, where both and are negative, the maximizers are quite different: when one of the parameters is and the other is suitably negative, maximality is proved for the vertices of a regular simplex or endpoints of an interval. Much more is proved in dimensions and , where for large regions of the third quadrant, maximizers are shown to consist of the vertices of intervals or equilateral triangles.

Paper Structure

This paper contains 13 sections, 18 theorems, 111 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Riesz vs. Riesz: capacity ratios in $1$ dimension
  3. Riesz vs. Riesz for $2$ dimensions
  4. Higher dimensions: results and open problems
  5. Examples for \ref{['conj:ps45']} and \ref{['conj:riesz']}
  6. Proof of \ref{['th:onedim']}: minimizing capacity for given length, in $1$ dimension
  7. Simplex examples, and proofs of \ref{['th:symmetrybreaking2dim']} and \ref{['th:trianglecapformula']}
  8. Proof of \ref{['th:threeptratio']}: optimizing capacity ratios amongst two- and three-point sets
  9. Proof of \ref{['th:2deqtriangle']}: planar sets when $p<0, q \leq-2$
  10. Convergence of Riesz capacity wrt Hausdorff distance
  11. Proof of \ref{['th:isodiametric']}: capacitary Brunn--Minkowski and isodiametric inequalities
  12. Additional proofs
  13. Potential theoretic facts

Key Result

Theorem 1

Suppose $K$ is a compact subset of ${{\mathbb R}^n}, n \geq 1$. (a) (Monotonicity) $\mathop{\mathrm{Cap_\mathit{p}}}\nolimits(K)$ is a decreasing function of $p \in {\mathbb R}$, and is strictly decreasing on the interval where it is positive. (b) (Diameter: $p = -\infty$) and $2^{1/p} \leq \mathop{\mathrm{Cap_\mathit{p}}}\nolimits(K)/\mathop{\mathrm{diam}}\nolimits(K) \leq 1$ for all $p <0$. If

Figures (6)

  • Figure 0: The Riesz capacity $\mathop{\mathrm{Cap_\mathit{p}}}\nolimits(\overline{{\mathbb B}}^n)$ of the closed unit ball, plotted as a function of $p$ for the first three values of $n$. The vertical intercept at $p=0$ is the logarithmic capacity $\mathop{\mathrm{Cap_0}}\nolimits(\overline{{\mathbb B}}^n)$, with values $1/2, 1, 2e^{-1/2}$ for $n=1,2,3$ respectively. When $p \leq -2$, the capacity is $2^{1+1/p}$ for each $n$, with limiting value $2$ (the diameter of the ball) as $p \to -\infty$. The plots are based on standard capacity formulas CL24b.
  • Figure 1: ($n=1$) Maximizing $\mathop{\mathrm{Cap_\mathit{q}}}\nolimits(K)/\mathop{\mathrm{Cap_\mathit{p}}}\nolimits(K)$ for $K \subset {\mathbb R}$. Solid regions and segments on the boundary indicate rigorous results. Striped regions are conjectural. Blue corresponds to the interval and green to the two-point set.
  • Figure 2: ($n=2$) Maximizing $\mathop{\mathrm{Cap_\mathit{q}}}\nolimits(K)/\mathop{\mathrm{Cap_\mathit{p}}}\nolimits(K)$ for $K \subset {\mathbb R}^2$. Solid regions and solid segments indicate rigorous results. Striped regions and dashed curves are conjectural. Blue corresponds to the disk, red to the regular three-point set, and green to the two-point set.
  • Figure 3: ($n \geq 3$) Maximizing $\mathop{\mathrm{Cap_\mathit{q}}}\nolimits(K)/\mathop{\mathrm{Cap_\mathit{p}}}\nolimits(K)$ for $K \subset {{\mathbb R}^n}$. Solid regions of the diagram and the solid segments indicate rigorous results. Striped regions and dashed curves are conjectural. Blue corresponds to the ball, red to the regular $(n+1)$-point set, and green to the two-point set. For dimensions $n=1$ and $n=2$, see the additional results in \ref{['fig:pqdiagram1D']} and \ref{['fig:pqdiagram2D']}.
  • Figure 4: Plot of $\mathop{\mathrm{Cap_\mathit{q}}}\nolimits(\overline{{\mathbb B}}^n)/\mathop{\mathrm{Cap_\mathit{p}}}\nolimits(\overline{{\mathbb B}}^n)$ as a function of $n$, for $p=2$ and $q=8$. The ratio increases with $n$, in accordance with \ref{['conj:riesz']}.
  • ...and 1 more figures

Theorems & Definitions (27)

  • Theorem 1: Clark and Laugesen CL24b
  • Theorem 2: Maximizing length divided by $p$-capacity, i.e. minimizing $p$-capacity for given length
  • Proposition 1: Maximizing $q$-capacity given diameter
  • Lemma 1
  • Proposition 2: Minimizing $p$-capacity for given diameter
  • Conjecture 1: Pólya and Szegő PS45
  • Theorem 3: Symmetry breaking
  • Theorem 4: Capacity of a three-point set
  • Theorem 5: Optimal capacity ratios among two- and three-point sets
  • Theorem 6: $p<0$ and $q\leq -2$
  • ...and 17 more