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 $ \left( \frac {t}{2} \right) w $, where $w$ is the distance between $L_1$ and $L_2$. What is the smallest $t$ such that \[ S \cap \bigcap_K B_S (K, t) \ne \emptyset \]for all $S$? ($K$ runs over all lines in the plane.)
**
   - Let \( A \) be a point on the boundary of \( S \).
   - Define \( S_A \) as the convex set homothetic to \( S \) with pole \( A \) and ratio \( \frac{2}{3} \).

8. **Support Lines:**
   - If \( L_1 \) and \( L_2 \) are two support lines parallel to \( K \) and \( A = L_1 \cap S \), \( A' = L_2 \cap S \), then \( S_A \cap S_{A'} \subset S \cap B_S(K, \frac{1}{3}) \).

9. **Intersection Argument:**
   - It is enough to prove \( \bigcap_{A \in \partial S} S_A \neq \emptyset \).
   - Suppose \( A, B, C \in \partial S \) and \( M \) is the centroid of \( \triangle ABC \).
   - Clearly, \( M \in S_A \cap S_B \cap S_C \).

10. **Helly's Theorem:**
    - By Helly's theorem, if \( N \) is a set consisting of a finite number of points on the boundary of \( S \), then \( \bigcap_{A \in N} S_A \neq \emptyset \).
    - The compactness of sets \( S_A \) ensures the intersection is non-empty.

\(\blacksquare\)

The final answer is \( \boxed{ t = \frac{1}{3} } \).