Convexity arises naturally in the plane and in space, as the preceding pages illustrate. In fact, it applies just as naturally in the n-dimensional space ℝ*n* and in more general vector spaces (including infinite-dimensional ones).
This notion is particularly rich in both properties and applications. What happens if we slightly alter the definitions of convexity? Does this lead to interesting new ideas?
To find out, let A be a nonempty subset of the plane or of our vector space.
Stars everywhere!
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To say that A is convex is to require that, for every P in A and every Q in A, the entire line segment [PQ] lies in A. Replace the first quantifier ("for every P in A") with "there exists a P in A." This gives a notion distinct from convexity. A set A satisfying "there exists a P in A such that, for every Q in A, the entire line segment [PQ] lies in A" is called star-shaped (or starshaped in English).
