This paper presents various functional-geometric aspects of the logarithmic Sobolev capacity ${\mathrm{Cap}}_{\log ,\gamma ,p}$ - a capacity generated by the logarithmic Sobolev space $W_p^{\log ,\gamma }$, where $(\gamma ,p)\in (0,\infty )\times [1,\infty )$. This new nonlinear-inhomogeneous-nontrivial capacity is closely related to the logarithmic Hausdorff capacity. Not only continuously sharp embedding property of $W_p^{\log ,\gamma }$ into certain logarithmic weighted Lebesgue space and the tracing inequalities are established, but also the logarithmic perimeter and the Lebesgue measure are utilized to: reformulate ${\mathrm{Cap}}_{\log ,\gamma ,p}$; find the first variation of the logarithmic perimeter via the corresponding logarithmic mean curvature; characterize the hypersurface of a constant logarithmic mean curvature.

本文介绍了对数 Sobolev 容量的各种功能几何方面 ${\mathrm{帽}}_{\mathrm{日志},\gamma ,磷}$ - 由对数 Sobolev 空间生成的容量 ${宽}_{磷}^{\mathrm{日志},\gamma }$， 在哪里 $(\gamma ,磷)\in (0,\infty )\times [1,\infty )$. 这种新的非线性非齐次非平凡容量与对数 Hausdorff 容量密切相关。不仅具有连续尖锐的嵌入特性${宽}_{磷}^{\mathrm{日志},\gamma }$ 进入某些对数加权 Lebesgue 空间并建立追踪不等式，但也利用对数周长和 Lebesgue 测度来： ${\mathrm{帽}}_{\mathrm{日志},\gamma ,磷}$; 通过相应的对数平均曲率找到对数周长的第一个变化；表征恒定对数平均曲率的超曲面。