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We prove a new logarithmic epiperimetric inequality for multiplicity-onestationary cones with isolated singularity by flowing in the radial directionany given trace along appropriately chosen directions. In contrast to previousepiperimetric inequalities for minimal surfaces (e.g. those of Reifenberg,Taylor and White), we need no a priori assumptions on the structure of the cone(e.g. integrability). Moreover, if the cone is integrable (not only throughrotations), we recover the classical epiperimetric inequality. As a consequencewe deduce a new $\varepsilon$-regularity result for almost area-minimizingcurrents at singular points, where at least one blow-up is a multiplicity-onecone with isolated singularity. This result is similar to the one forstationary varifolds of Leon Simon, but independent from it since almostminimizers do not satisfy any equation.

(Log-)epiperimetric inequality and regularity over smooth cones for almost Area-Minimizing currents