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Let V be an origin-symmetric convex body in ${ℝ}^n$, n≥ 2, of Gaussian measure ${\gamma }_n\left(V\right)\ge 1/2$. It is proved that for every choice $u_1,...,u_n$ of vectors in the Euclidean unit ball $B_n$, there exist signs ${\epsilon }_j\in -1,1$ with ${\epsilon }_1{u}_1+...+{\epsilon }_n{u}_n\in \left(clogn\right)V$. The method used can be modified to give simple proofs of several related results of J. Spencer and E. D. Gluskin.