Putnam 1990 B6

Let $S$ be a nonempty closed bounded convex set in the plane. Let $K$ be a line and $t$ a positive number. Let $L_1$ and $L_2$ be support lines for $S$ parallel to $K_1$, and let $\overline{L}$ be the line parallel to $K$ and midway between $L_1$ and $L_2$. Let $B_S(K, t)$ be the band of points whose distance from $\overline{L}$ is at most $(t/2)w$, where $w$ is the distance between $L_1$ and $L_2$. What is the smallest $t$ such that $$S \cap \bigcap_K B_S(K, t) \neq \emptyset$$ for all $S$? ($K$ runs over all lines in the plane.)