Suppose I have some n-dimensional point
$ \mathbf{x} = [x_1,...,x_n] $
where some of the $x_i$ have different units (for example, $x_1$ in Ohms and $x_5$ in seconds). Now if all the elements had the same units I could define a ball $B$ of radius $r$ around $\mathbf{x}$ by
$ B_r(\mathbf{x}) = \{ \mathbf{y} : ||\mathbf{x}-\mathbf{y}|| \leq r\}$
But because the elements of the point do not have the same units, the norm $||\mathbf{x}-\mathbf{y}||$ is not meaningful if it is taken to be the Euclidean norm or something similar.
How could one define a meaningful distance in this space in order to define a ball of a particular radius around a point? My first approach is to consider that each element $x_i$ is bounded by some interval with a minimum $m_i$ and a maximum $M_i$ such that
$x_i \in [m_i, M_i]$
and then define a radius for each element by
$r^*_i = \frac{1}{(M_i-m_i)}r$
Then I could define a "ball" of points around the point in question by
$ B_r(\mathbf{x}) = \{ \mathbf{y} : |x_i - y_i| \leq r^*_i\}$
But I am not sure that this is meaningful. Ultimately I am trying to define a ball of parameters for a dynamical model around a particular parameter, and the model is to be evaluated at each point of this ball (a "parameter hypersphere" instead of a parameter grid). Any help here is appreciated.
Any metric formed in the way you speak of will be topologically equivalent to the standard Euclidean one whose coordinates will be your 'meaningless' Ohms-seconds-etc. This is because in finite dimensions over $\mathbb{R}$ the normal Euclidean metric is equivalent to the 'taxi-cab metric' which considers the sums of the distances in each variable. Scaling each one-dimensional distance by the (presumed non-zero) factors $(M_i - m_i)^{-1}$ will again give you an equivalent metric. This is a basic exercise in topology: You can easily show that for any point, in either metric you can find neighborhoods (e.g. n-dimensional rectangular prisms centered around the point with side lengths converging to zero) that are bases for the different topologies but which are nested in each other.