I'm wondering whether there exists a geometric analog concept of absolute value. In other words, if absolute value can be defined as
$$ \text{abs}(x) =\max(x,-x) $$
intuitively the additive distance from $0$ to $x$, is there a geometric version
$$ \text{Geoabs}(x) = \max(x, 1/x) $$
which is intuitively the multiplicative "distance" from $1$ to $x$?
Update: Agreed it only makes sense for $Geoabs()$ to be restricted to positive reals.
To give some context on application, I am working on the solution of an optimization problem something like:
$$ \begin{array}{ll} \text{minimize} & \prod_i Geoabs(x_i) \\ \text{subject to} & \prod_{i \in S_j} x_i = C_j && \forall j \\ &x_i > 0 && \forall i . \end{array} $$
Basically want to satisfy all these product equations $j$ by moving $x_i$'s as little as possible from $1$. Note by the construction there are always infinite feasible solutions.
from Hot Weekly Questions - Mathematics Stack Exchange
dashnick
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