ÖMO 1991 Problem 1

Consider a convex solid $K$ in space and two parallel planes $\epsilon _1$ and $\epsilon _2$ on the distance $1$ tangent to $K$ . A plane $\epsilon$ between $\epsilon _1$ and $\epsilon _2$ is on the distance $d_1$ from $\epsilon _1$ . Find all $d_1$ such that the part of $K$ between $\epsilon _1$ and $\epsilon$ always has a volume not exceeding half the volume of $K$ .