R. Y. Liu [Ann. Stat. 18, No. 1, 405-414 (1990; Zbl 0701.62063)] introduced the simplicial median as a generalization of the median for a multivariate probability distribution. The simplicial median is affine equivariant and so is the sample simplicial median if it is unique or if it is conveniently chosen when it is not unique. The object of this article is to prove that Liu’s sample simplicial median and its modified version [M. A. Arcones and E. Giné, Ann. Probab. 21, No. 3, 1494-1542 (1993; Zbl 0789.60031)] are both robust. Their robustness will be measured by means of the finite sample breakdown point.