Proof

(related to Corollary: Intersection of Convex Affine Sets)

Let \(\mathcal C\) be a family of convex subsets \(C\) of an \(n\)-dimensional affine space \(\mathcal A=(A,V_A,v)\) with \(V_A\) as the associated vector space over the field \(\mathbb R\) of real numbers. Take two points \(P,Q\) from the intersection \(\bigcap\mathcal C\).

By definition of set intersection, for all \(C\in\mathcal C\) we have that \(P,Q\in C\).

The convexity of each C yields that the straight line \(\lambda P+(1-\lambda)Q\) is contained in each \(C\).

Therefore, these elements are also in the intersection \(\bigcap\mathcal C\), by definition of set intersection.

Hence \(\bigcap\mathcal C\) is also convex.


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References

Bibliography

  1. Ziegler, G√ľnter M.: "Lectures on Polytopes", Springer Verlag, 1994