Uniform directional margin inside a cone

Let $\mathcal C\subset\mathbb R^n\setminus\{0\}$ be open and invariant under positive scaling. A family of nonzero vectors $T(x)$ has a uniform directional margin in $\mathcal C$ if

\operatorname{dist}\!\left(\frac{T(x)}{|T(x)|},\mathbb R^n\setminus\mathcal C\right)\ge\delta>0

for every parameter $x$ . The condition concerns the unit direction, not a lower bound on $|T(x)|$ .

Compactness and vanishing amplitudes

If the normalized direction extends continuously to a compact parameter set with values in $\mathcal C$ , its distance from the closed complement has a positive minimum. Hence the margin can persist while $T$ tends to zero at a boundary. A relative perturbation $|\Delta T|\le\varepsilon|T|$ , for sufficiently small $\varepsilon$ depending on the margin, keeps the perturbed vector inside the cone. An absolute perturbation bound alone does not give this conclusion near zero.