Directional Lipschitzness of minimal time functions in Hausdorff topological vector spaces (Q456994)

Let \(E\) be a general Hausdorff topological vector space and let \(T_{S,\Omega}: E\to\overline{\mathbb{R}}_+\) be defined by \[ T_{S,\Omega}(x):= \text{inf}\{t> 0: S\cap(x+ t\Omega)\neq \varnothing\} \] be the so-called minimal time function with respect to the subsets \(S\) and \(\Omega\) of \(E\). Obviously, this function is an extension of the three crucial functions in variational analysis: the indicator function, the distance function and the Minkowsky function. In a former paper, the author studied the directional derivatives, the subdifferentials and the regularities of \(T_{S,\Omega}\). In the present paper, he continues this investigations discussing such closed sets \(S\subset E\) for which the function \(T_{S,\Omega}\) (for a given bounded closed set \(\Omega\subset E\)) is directionally Lipschitz in the sense of Rockafellar. The characterization of these sets (the so-called \(\Omega\)-epi-Lipschitz sets) is realized in terms of new (the \(\Omega\)-hypertangent cone, the \(\Omega\)-quasi-hypertangent cone and the \(\Omega\)-Clarke tangent cone) and classical (the Bouligand tangent cone) cone approximations. All presented assertions are generalizations of known results regarding epi-Lipschitz sets, compactly epi-Lipschitz sets and \(K\)-directionally Lipschitz sets.