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Let $(X, d)$ be a metric space. We will say that $\gamma \colon [a,b] \to X$ is of bounded variation, if \begin{equation} V(\gamma) = \sup_{a=t_0 < \cdots < t_n < b} \sum_{i=1}^n d( \gamma(t_i), \gamma(t_{i-1})) \end{equation} satisfies $V(\gamma) < \infty$. The variation $V(\gamma)$ can be thought of as a generalization of a length of a curve.

Now, let's say that we have function $c \colon \times_{i=1}^n [a_i, b_i] \to X$. Is there some analogous generalization of $n$-dimensional "volume", or just, when $n=2$, some generalization of area, for such a function $c$? I'm especially interested in ones that would not require $X$ to be endowed with some additional structure.

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Just a remark, too long for a comment. For quite a long time, $BV$ functions were studied only in the case of functions defined on the real line. Then there were some attempts to get a relevant definition on $\mathbb R^n$ or on an open subset of $\mathbb R^n$, following the type of definition that you suggest: for some time, it was not clear what was the proper and relevant definition, and the situation was clarified with the popularization of Distribution Theory: a function $f:\mathbb R^n\longrightarrow \mathbb R$ is $BV$ means that $f$ belongs to $L^1$ and its distribution derivative is a bounded Radon measure: it is then possible to apply the (isoperimetric) Gagliardo-Nirenberg Inequality and to prove that $f$ belongs to $L^\frac{n}{n-1}(\mathbb R^n)$.

Federer, in his celebrated and hard-to-read book used extensively the $BV$ regularity to tackle Stokes formulas for open subsets $\Omega$ whose perimeter is finite, meaning that the indicatrix function of $\Omega$ is of bounded variation. It is my opinion that if you are eager with generalizations of $BV$ regularity, you should start exploring the known cases, some of which are studied in Federer's book.

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  • $\begingroup$ As a comment on that comment, especially in the context of Federer, $BV$ can be considered a special case of normal currents. Ambrosio and Kirchheim then later extended those to metric spaces, though not in the precise way asked for in the question. The general problem is that it is not a-priori clear what the volume of a jump is if there is no simple linear interpolation between points. The idea of cartesian currents (Giaquinta, Modica & Soucek) should provide a way, but afaik has never been generalized to metric spaces. So I am not sure if there is a way to answer without writing 2-3 papers. $\endgroup$
    – mlk
    Commented Apr 13, 2023 at 8:21

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