  Understanding the stochastic behavior of random projections of geometric sets
constitutes a fundamental problem in high dimension probability that finds wide
applications in diverse fields. This paper provides a kinematic description for
the behavior of Gaussian random projections of closed convex cones, in analogy
to that of randomly rotated cones studied in [ALMT14]. Formally, let $K$ be a
closed convex cone in $\mathbb{R}^n$, and $G\in \mathbb{R}^{m\times n}$ be a
Gaussian matrix with i.i.d. $\mathcal{N}(0,1)$ entries. We show that $GK\equiv
\{G\mu: \mu \in K\}$ behaves like a randomly rotated cone in $\mathbb{R}^m$
with statistical dimension $\min\{\delta(K),m\}$, in the following kinematic
sense: for any fixed closed convex cone $L$ in $\mathbb{R}^m$, \begin{align*}
&\delta(L)+\delta(K)\ll m\, \Rightarrow\, L\cap GK = \{0\} \hbox{ with high
probability},\\ &\delta(L)+\delta(K)\gg m\, \Rightarrow\, L\cap GK \neq \{0\}
\hbox{ with high probability}. \end{align*} A similar kinematic description is
obtained for $G^{-1}L\equiv \{\mu \in \mathbb{R}^n: G\mu \in L\}$.
  The practical usefulness and broad applicability of the prescribed
approximate kinematic formulae are demonstrated in a number of distinct
problems arising from statistical learning, mathematical programming and
asymptotic geometric analysis. In particular, we prove (i) new phase
transitions of the existence of cone constrained maximum likelihood estimators
in logistic regression, (ii) new phase transitions of the cost optimum of
deterministic conic programs with random constraints, and (iii) a local version
of the Gaussian Dvoretzky-Milman theorem that describes almost deterministic,
low-dimensional behaviors of subspace sections of randomly projected convex
sets.
