Differentiating maps into \(L^1\), and the geometry of BV functions. (English) Zbl 1194.22009

Considering a metric measure space \(X\) and a Banach space \(V\) the authors examine the connection between differentiation theory for Lipschitz maps \(X\to V\) and bi-Lipschitz non-embeddability. The paper is devoted to the case when \(V=L^1\), where the differentiability fails. Supposing that the space \(X\) carries a measure, the authors give some alternative characterizations of \(L^1\) maps \(f\to L^1\) and discuss the equivalence between metrics induced by maps to \(L^1\) and cut metrics. The total perimeter measure \(\lambda\in \text{Radon}(X)\) associated to a finite perimeter cut measure is constructed and its properties are studied taking into account location and scale.
The notion of bad part of \(\lambda\) is also defined and it is the key in proving the main differentiation theorem given in this paper. A new type of differentiability is established for certain \(X\), including \(R^n\) and the Heisenberg group \(H\) with its Carnot-Carathéodory metric. It is shown that \(H\) does not bi-Lipschitz embed into \(L^1\). This result provides a natural counterexample to Goemans-Linial conjecture in theoretical computer science. The new connection established between the Lipschitz maps to \(L^1\) and the functions of bounded variation (BV) allows to exploit different results on the structure of BV functions on the Heisenberg group.


22E30 Analysis on real and complex Lie groups
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
46B20 Geometry and structure of normed linear spaces
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
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