Colloquial Math - Convexity
Today a student asked Edward:
Can you please briefly give me a more general notion of convexity?
I’ve studied that in 2D a function is convex if, considering 2 points on it, the function stays under the segment that connects them but is there any more general formulation for this concept ?
Edward:
A more general definition could be formulated considering a $ L $ Vector Space and $ V \subset L $ a subspace.
Given 2 any points $ x_{1}, x_{2} \in V $ it is said that $ V $ is convex if the linear parametrization
$$ s(t) = t x_{1} + (1 - t) x_{2} \qquad t \in (0,1) $$
brings to $ s(t) \in V \quad \forall t \in (0,1) $ so the path stays alwyas inside the Space.
Hence a convex space is closed wrt any linear path connecting any 2 elements in it.
This kind of definition connects to the one known by the students because more generally in $ \mathbb{R}^{n} $ a general $ n-1 $ dimensional space, divides the container space in 2 subspaces and one of it could be convex wrt the above definition.
