  We consider the problem of estimating small ball probabilities $\mathbb
P\{f(G) \leqslant \delta \mathbb Ef(G)\}$ for sub-additive,positively
homogeneous functions $f$ with respect to the Gaussian measure. We establish
estimates that depend on global parameters of the underlying function which
take into account analytic and statistical measures, such as the variance and
the $L^1$-norms of its partial derivatives. This leads to dimension-dependent
bounds for small ball and lower small deviation estimates for seminorms when
the linear structure is appropriately chosen to optimize the aforementioned
parameters. Our bounds are best possible up to numerical constants. In all
regimes, $\|G\|_\infty = \max_{ i \leqslant n}|g_i|$ arises as an extremal case
in this study. The proofs exploit the convexity and hypercontractivity
properties of the Gaussian measure.
