The meta-question is: understanding by how much a set fails to have full capacity. I will pin it down to some concrete, although non-exhaustive, questions in a reasonably simple framework. Let ${\mathbb D}_r$ be the disc in the plane centered at the origin and having radius $r\in(0,1)$ and ${\mathbb S}=\partial{\mathbb D}_1$. Consider subsets $E=\cup_i I_i\subseteq{\mathbb S}$ which are finite unions of disjoint closed arcs and define the following quantities: $$ \text{Cap}_1(E)=\inf\left\{\int_{{\mathbb D}_1}|\nabla u|^2dxdy:\ u|_{{\mathbb D}_{1/2}}=0,\ u|_E\ge1\right\}; $$ $$ \text{Cap}_2(E)=\sup\left\{\frac{\mu(E)^2}{\int_0^{2\pi}\left(\int_0^{2\pi}\frac{d\mu(s)}{|t-s|^{1/2}}\right)^2d\mu(t)}:\ \text{supp}(\mu)\subseteq E\right\}; $$ $$ \text{Cap}_3(E)=\sup\left\{\frac{\mu(E)^2}{\int_0^{2\pi}\int_0^{2\pi}\log\frac{2}{|e^{is}-e^{it}|}d\mu(s)d\mu(t)}:\ \text{supp}(\mu)\subseteq E\right\}. $$ These are three avatars of logarithmic capacity, this meaning that $$ \text{Cap}_1(E)\approx\text{Cap}_2(E)\approx\text{Cap}_3(E). $$ (Here $\approx$ means that the term on the left is bounded from below and above by a positive multiple of the term on the right).

There are many other quantities that are equivalent to $\text{Cap}_j(E)$ and which involve Green's functions, holomorphic functions, stochastic processes, dyadic objects...

The question is: which equivalences hold for the capacity deficit? For instance, is it true that: $$ \text{Cap}_1({\mathbb S})-\text{Cap}_2(E)\approx\text{Cap}_2({\mathbb S})-\text{Cap}_1(E), $$ or that $$ \text{Cap}_2({\mathbb S})-\text{Cap}_2(E)\approx\text{Cap}_3({\mathbb S})-\text{Cap}_3(E)? $$ I have been thinking on and off on this for a few years. The motivation is that, in the dyadic case, some estimates for the capacity of a condenser depend on this capacity deficit and a better understanding of it might lead to the extension of the estimates to classical Potential Theory. Here is a reference: https://arxiv.org/abs/1108.5325. For the equivalence of dyadic and classical capacity see: Benjamini, Itai; Peres, Yuval, Random walks on a tree and capacity in the interval. Ann. Inst. H. Poincaré Probab. Statist. 28 (1992), no. 4, 557–592.

The meta-question is: understanding by how much a set fails to have full capacity. I will pin it down to some concrete, although non-exhaustive, questions in a reasonably simple framework. Let ${\mathbb D}_r$ be the disc in the plane centered at the origin and having radius $r\in(0,1)$ and ${\mathbb S}=\partial{\mathbb D}_1$. Consider subsets $E=\cup_i I_i\subseteq{\mathbb S}$ which are finite unions of disjoint closed arcs and define the following quantities: $$ \text{Cap}_1(E)=\inf\left\{\int_{{\mathbb D}_1}|\nabla u|^2dxdy:\ u|_{{\mathbb D}_{1/2}}=0,\ u|_E\ge1\right\}; $$ $$ \text{Cap}_2(E)=\sup\left\{\frac{\mu(E)^2}{\int_0^{2\pi}\left(\int_0^{2\pi}\frac{d\mu(s)}{|t-s|^{1/2}}\right)^2d\mu(t)}:\ \text{supp}(\mu)\subseteq E\right\}; $$ $$ \text{Cap}_3(E)=\sup\left\{\frac{\mu(E)^2}{\int_0^{2\pi}\int_0^{2\pi}\log\frac{2}{|e^{is}-e^{it}|}d\mu(s)d\mu(t)}:\ \text{supp}(\mu)\subseteq E\right\}. $$ These are three avatars of logarithmic capacity, this meaning that $$ \text{Cap}_1(E)\approx\text{Cap}_2(E)\approx\text{Cap}_3(E). $$ (Here $\approx$ means that the term on the left is bounded from below and above by a positive multiple of the term on the right).

There are many other quantities that are equivalent to $\text{Cap}_j(E)$ and which involve Green's functions, holomorphic functions, stochastic processes, dyadic objects...

The question is: which equivalences hold for the capacity deficit? For instance, is it true that: $$ \text{Cap}_1({\mathbb S})-\text{Cap}_1(E)\approx\text{Cap}_2({\mathbb S})-\text{Cap}_2(E), $$ or that $$ \text{Cap}_2({\mathbb S})-\text{Cap}_2(E)\approx\text{Cap}_3({\mathbb S})-\text{Cap}_3(E)? $$ I have been thinking on and off on this for a few years. The motivation is that, in the dyadic case, some estimates for the capacity of a condenser depend on this capacity deficit and a better understanding of it might lead to the extension of the estimates to classical Potential Theory. Here is a reference: https://arxiv.org/abs/1108.5325. For the equivalence of dyadic and classical capacity see: Benjamini, Itai; Peres, Yuval, Random walks on a tree and capacity in the interval. Ann. Inst. H. Poincaré Probab. Statist. 28 (1992), no. 4, 557–592.

The meta-question is: understanding how much a set fails to have full capacity. I will pin it down to some concrete, although non-exhaustive, questions in a reasonably simple framework. Let ${\mathbb D}_r$ be the disc in the plane centered at the origin and having radius $r\in(0,1)$ and ${\mathbb S}=\partial{\mathbb D}_1$. Consider subsets $E=\cup_i I_i\subseteq{\mathbb S}$ which are finite unions of disjoint closed arcs and define the following quantities: $$ \text{Cap}_1(E)=\inf\left\{\int_{{\mathbb D}_1}|\nabla u|^2dxdy:\ u|_{{\mathbb D}_{1/2}}=0,\ u|_E\ge1\right\}; $$ $$ \text{Cap}_2(E)=\sup\left\{\frac{\mu(E)^2}{\int_0^{2\pi}\left(\int_0^{2\pi}\frac{d\mu(s)}{|t-s|^{1/2}}\right)^2d\mu(t)}:\ \text{supp}(\mu)\subseteq E\right\}; $$ $$ \text{Cap}_3(E)=\sup\left\{\frac{\mu(E)^2}{\int_0^{2\pi}\int_0^{2\pi}\log\frac{2}{|e^{is}-e^{it}|}d\mu(s)d\mu(t)}:\ \text{supp}(\mu)\subseteq E\right\}. $$ These are three avatars of logarithmic capacity, this meaning that $$ \text{Cap}_1(E)\approx\text{Cap}_2(E)\approx\text{Cap}_3(E). $$ (Here $\approx$ means that the term on the left is bounded from below and above by a positive multiple of the term on the right).

There are many other quantities that are equivalent to $\text{Cap}_j(E)$ and which involve Green's functions, holomorphic functions, stochastic processes, dyadic objects...

The question is: which equivalences hold for the capacity deficit? For instance, is it true that: $$ \text{Cap}_1({\mathbb S})-\text{Cap}_2(E)\approx\text{Cap}_2({\mathbb S})-\text{Cap}_1(E), $$ or that $$ \text{Cap}_2({\mathbb S})-\text{Cap}_2(E)\approx\text{Cap}_3({\mathbb S})-\text{Cap}_3(E)? $$ I have been thinking on and off on this for a few years. The motivation is that, in the dyadic case, some estimates for the capacity of a condenser depend on this capacity deficit and a better understanding of it might lead to the extension of the estimates to classical Potential Theory. Here is a reference: https://arxiv.org/abs/1108.5325. For the equivalence of dyadic and classical capacity see: Benjamini, Itai; Peres, Yuval, Random walks on a tree and capacity in the interval. Ann. Inst. H. Poincaré Probab. Statist. 28 (1992), no. 4, 557–592.

The meta-question is: understanding by how much a set fails to have full capacity. I will pin it down to some concrete, although non-exhaustive, questions in a reasonably simple framework. Let ${\mathbb D}_r$ be the disc in the plane centered at the origin and having radius $r\in(0,1)$ and ${\mathbb S}=\partial{\mathbb D}_1$. Consider subsets $E=\cup_i I_i\subseteq{\mathbb S}$ which are finite unions of disjoint closed arcs and define the following quantities: $$ \text{Cap}_1(E)=\inf\left\{\int_{{\mathbb D}_1}|\nabla u|^2dxdy:\ u|_{{\mathbb D}_{1/2}}=0,\ u|_E\ge1\right\}; $$ $$ \text{Cap}_2(E)=\sup\left\{\frac{\mu(E)^2}{\int_0^{2\pi}\left(\int_0^{2\pi}\frac{d\mu(s)}{|t-s|^{1/2}}\right)^2d\mu(t)}:\ \text{supp}(\mu)\subseteq E\right\}; $$ $$ \text{Cap}_3(E)=\sup\left\{\frac{\mu(E)^2}{\int_0^{2\pi}\int_0^{2\pi}\log\frac{2}{|e^{is}-e^{it}|}d\mu(s)d\mu(t)}:\ \text{supp}(\mu)\subseteq E\right\}. $$ These are three avatars of logarithmic capacity, this meaning that $$ \text{Cap}_1(E)\approx\text{Cap}_2(E)\approx\text{Cap}_3(E). $$ (Here $\approx$ means that the term on the left is bounded from below and above by a positive multiple of the term on the right).

There are many other quantities that are equivalent to $\text{Cap}_j(E)$ and which involve Green's functions, holomorphic functions, stochastic processes, dyadic objects...

The question is: which equivalences hold for the capacity deficit? For instance, is it true that: $$ \text{Cap}_1({\mathbb S})-\text{Cap}_2(E)\approx\text{Cap}_2({\mathbb S})-\text{Cap}_1(E), $$ or that $$ \text{Cap}_2({\mathbb S})-\text{Cap}_2(E)\approx\text{Cap}_3({\mathbb S})-\text{Cap}_3(E)? $$ I have been thinking on and off on this for a few years. The motivation is that, in the dyadic case, some estimates for the capacity of a condenser depend on this capacity deficit and a better understanding of it might lead to the extension of the estimates to classical Potential Theory. Here is a reference: https://arxiv.org/abs/1108.5325. For the equivalence of dyadic and classical capacity see: Benjamini, Itai; Peres, Yuval, Random walks on a tree and capacity in the interval. Ann. Inst. H. Poincaré Probab. Statist. 28 (1992), no. 4, 557–592.