Let \(F\) be a compact convex set of diameter 1 in Euclidean \(n\)-space. The author proves that if \(F\) contains an \((n-1)\)-dimensional regular simplex with edge 1, then \(F\) can be partitioned into \(n + 1\) sets of a diameter smaller than 1. She shows a few other conditions under which \(F\) can be partitioned into \(n + 1\) sets of a smaller diameter. Also the following theorem is proved: if every unit segment of \(F\) has exactly one common point with a fixed \(k\)-dimensional plane, where \(0 \leq k \leq n- 1\), then \(F\) can be partitioned into \(n - k + 1\) sets of a diameter smaller than 1.