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Helly -> Centring -> Blaschke

Prove that if a finite collection of convex sets in the plane has the property that every three sets from the collection have a point in common, then they all have a point in common. Prove that this holds for an infinite collection also if the sets from the given collection are bounded.

Consider a closed convex figure in plane. Prove that we can find an interior point $O$ such that for any chord $PQ$ of the figure passing through $O$ we have $PO,OQ >\displaystyle \frac{1}{3}PQ$.

The breadth of a closed convex curve in a given direction is the distance oh the two parallel lines of the given direction which enclose the given curve. The width of a curve the minimum value of the breadth as the direction varies. Prove that a convex curve of width $1$ contains a circle of radius $1/3$, and this is the best value possible.
Reference: J.H Cadwell, Topics in Recreational Mathematics

Every one of the problems above can be solved using the previous one.
For Helly’s Theorem see this link.

For the second problem, choose any three points on the boundary of our convex figure $F$, and name them $A,B,C$, respectively. Then consider the homotety $h_X$ of center $X$ and ratio $2/3$, where $X \in \{A,B,C\}$. Then the centroid of the triangle $ABC$ will be in the intersection $h_A(F)\cap h_B(F)\cap h_C(F)$. Helly’s theorem implies that $\displaystyle \bigcap_{X \in bd(F)} h_X(F) \neq \emptyset$. Choose $O$ in this intersection and we are done.

For the third problem, the equilateral triangle having heights equal to $1$ provides the case when there is no better value than $1/3$. In a convex figure with breadth $1$ choose a point $O$ like in the previous problem and by its properties we are done.

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