We study geodesics for plurisubharmonic functions from the Cegrell class${\mathcal F}_1$ on a bounded hyperconvex domain of ${\mathbb C}^n$ and showthat, as in the case of metrics on K\"{a}hler compact menifolds, they linearizean energy functional. As a consequence, we get a uniqueness theorem forfunctions from ${\mathcal F}_1$ in terms of total masses of certain mixedMonge-Amp\`ere currents. Geodesics of relative extremal functions areconsidered and a reverse Brunn-Minkowski inequality is proved for capacities ofmultiplicative combinations of multi-circled compact sets. We also show thatfunctions with strong singularities generally cannot be connected by(sub)geodesic arks.
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