Let $(X,d)$ be a compact metric space. Assume that $f:X\to \mathbb{R}$ is a positive continuous function. We say that the $f$-dimension of $(X,d)$ is equal to $0$ if for every $\epsilon>0$ there exists a finite open cover $D_i,\;i=1,2,\ldots n $ consists of open discs $D_i$ centered at $x_i$ such that $$ \sum_{i=1}^n (diam D_i)^{f(x_i)} \leq \epsilon,\; \forall t_i\in D_i $$

This situation is denoted by $\mathcal{H}^f(X)=0$

We define the generalized Hausdorff dimension of $X$ with $$ dim(X)=\inf \{f\mid \mathcal{H}^f(X)=0\} $$

So the dimension is no longer a number but a lower semi-continuous function $f$.

For what kind of metric space is this dimension $f$ a continuous function? For what kind of metric spaces is this dimension a constant function (equal to its Hausdorff dimension)?

Let $(X,d)$ be a compact metric space. Assume that $f:X\to \mathbb{R}$ is a positive continuous function. We say that the $f$-dimension of $(X,d)$ is equal to $0$ if for every $\epsilon>0$ there exists a finite open cover $D_i,\;i=1,2,\ldots n $ consists of open discs $D_i$ such that $$ \sum_{i=1}^n (diam D_i)^{f(t_i)} \leq \epsilon,\; \forall t_i\in D_i $$

This situation is denoted by $\mathcal{H}^f(X)=0$

We define the generalized Hausdorff dimension of $X$ with $$ dim(X)=\inf \{f\mid \mathcal{H}^f(X)=0\} $$

So the dimension is no longer a number but a lower semi-continuous function $f$.

For what kind of metric space is this dimension $f$ a continuous function? For what kind of metric spaces is this dimension a constant function (equal to its Hausdorff dimension)?

Let $(X,d)$ be a compact metric space. Assume that $f:X\to \mathbb{R}$ is a positive continuous function. We say that the $f$-dimension of $(X,d)$ is equal to $0$ if for every $\epsilon>0$ there exists a finite open cover $D_i,\;i=1,2,\ldots n $ consists of open discs $D_i$ such that $$\sum_{i=1}^n (diam D_i)^{f(t_i)} \leq \epsilon,\; \forall t_i\in D_i$$

This situation is denoted by $\mathcal{H}^f(X)=0$

We define the generalized Hausdorff dimension of $X$ with $$dim(X)=\inf \{f\mid \mathcal{H}^f(X)=0\}$$

So the dimension is no longer a number but is a lower semi continuous.

For what kind of metric space this dimension is a continuous function? For what kind of metric spaces this dimension is a constant function(equal to its Hausdorff dimension)?

Let $(X,d)$ be a compact metric space. Assume that $f:X\to \mathbb{R}$ is a positive continuous function. We say that the $f$-dimension of $(X,d)$ is equal to $0$ if for every $\epsilon>0$ there exists a finite open cover $D_i,\;i=1,2,\ldots n $ consists of open discs $D_i$ centered at $x_i$ such that $$ \sum_{i=1}^n (diam D_i)^{f(x_i)} \leq \epsilon,\; \forall t_i\in D_i $$

This situation is denoted by $\mathcal{H}^f(X)=0$

We define the generalized Hausdorff dimension of $X$ with $$ dim(X)=\inf \{f\mid \mathcal{H}^f(X)=0\} $$

So the dimension is no longer a number but a lower semi-continuous function $f$.

For what kind of metric space is this dimension $f$ a continuous function? For what kind of metric spaces is this dimension a constant function  (equal to its Hausdorff dimension)?