This paper supplements a result by \textit{R. Hettich} and \textit{P. Zencke} [in: Konstruktive Methoden der finiten nichtlinearen Optimierung, Tag., Oberwolfach 1980, ISNM 55, 103-119 (1980; Zbl 0442.65051)] on strong unicity in semi-infinite optimization. For linear problems which have a solution and satisfy the Slater condition a characterization of strong unicity is given which implies the well known result that for linear Chebyshev approximation problems Haar's condition is necessary and sufficient for strong uniqueness. For the case of one-sided \(L_ 1\)-approximation a characterization of strong unicity is given from which a result by \textit{H. Strauss} [Numer. Math. 40, 229-243 (1982; Zbl 0492.41051)] on strong unicity can be derived. Finally, a result on the existence of a quadrature formula and a function that has more than one best one-sided \(L_ 1\)-approximation are presented. The proofs and additional material are announced to appear elsewhere.