is a geometric notion originally introduced for point clouds in [@BinYu] and extended here for arbitrary probability distributions.

\[def:contcones\] Let $\sigma \in (0,  \pi /4 {\normalcolor})$, $ \delta \in [0,1)$, and $r >0$. We say that the probability measure $\mu \in \mathcal{P}({\mathbb{R}}^k)$ has an orthogonal cone structure with parameters $(\sigma, \delta, r)$ if there exists an orthonormal basis for ${\mathbb{R}}^k$, $e_1, \dots, e_k$, such that $$\mu \left( \bigcup_{j=1}^{k} C(e_j, \sigma,r)     \right) \geq 1 - \delta,$$ where $C(e_j, \sigma,r)$ is the set $$C(e_j , \sigma,r) := \left\{  z \in {\mathbb{R}}^k \:  : \:      \frac{ z \cdot e_j}{|z|} > \cos(\sigma), \quad  |z| > r       \right\}.$$

In simple words, a measure $\mu \in \mathcal{P}({\mathbb{R}}^k)$ has an orthogonal cone structure if it is concentrated on cones centered at vectors which form an orthonormal basis for ${\mathbb{R}}^k$. The parameter $\sigma$ is the opening angle of the cones, $r$ is the radius of the ball centered at the origin that we remove from the cones to form the sets $C(e_j,\sigma,r)$, and $\delta$ is the amount of mass that is allowed to lie outside the cones. The condition $\sigma<\pi/4$ ensures that the sets $C(e_j,\sigma,r)$ do not overlap.

We are now ready to state our first main result.

