Let C be a centrally symmetric convex body in \({\mathbb{R}}^ 3\) and let \(p_ C(x)\), \(x\in S^ 2\), denote the perimeter of the orthogonal projection of C into a plane orthogonal to x. It is known that C is uniquely determined by \(p_ C(.)\), see e.g. K. J. Falconer, Am. Math. Mon. 90, 690-693 (1983; Zbl 0529.52001)]. The author gives a satisfying answer to the stability problem whether the reconstruction of C from p(.) depends continuously on \(p_ C(.)\) showing that the ’distance’ of two convex bodies C, D is bounded above by a certain expression involving the ’distance’ of \(p_ C(.)\), \(p_ D(.)\). The corresponding problem where the perimeter is replaced by the area is discussed for convex bodies of revolution. For related stability problems in convexity cf. Yu. E. Anikonov and V. N. Stepanov [Math. USSR, Sb. 44, 483-490 (1983; Zbl 0507.53041)], V. I. Diskant [Sib. Math. J. 14(1973), 466-469 (1974; Zbl 0281.52004)] and R. Schneider [Proc. Am. Math. Soc. 50, 365-368 (1975; Zbl 0339.52002)].