A new condition that ensures the equality \[ (1)\quad T_{K\cap L}(x)=T_ K(x)\cap T_ L(x),\quad x\in K\cap L, \] \[ T_ K(x)=\{v\in X| \quad \liminf_{h\downarrow 0+}[dist(K,x+hv)/h]=0\} \] for convex closed subsets K, L of a Hilbert space X is established. Using support functions of convex sets it is proved that the following relation implies (1): \(\exists c>0:\forall e\in X^*\sigma_{L\cap K}(e)=\inf \{\sigma_ L(e-e')+\sigma_ K(e')|\) \(e'\in X^*\), \(\| e'\|_{X^*}\leq c\| e\|_{X^*}\}\), \(\sigma_ K(e)=\sup_{x\in K}\), \(e\in X^*\).