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I'm struggling to understand an example we were given. The problem description is:

Let $f$ be a convex function in $E^n$. Prove that the set of subgradients of $f$ in a given point form a ... convex set.

I have the solution, but don't really understand it. According to my intuition, the set of subgradients form a non-convex set of the points in areas marked $y$ in the following picture:

enter image description here

What am I missing here? Thanks.

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Your intuition is wrong. Do you know the definition of subgradient?

In your picture, the subgradients are not the points in the region of the plane marked $y$, they are the slopes of the lines through $(x_0, f(x_0))$ in that region.

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    The "objects" are the slopes of the lines. The convexity is due to the fact that the constraints on the slopes are linear inequalities, and the solution sets of linear inequalities are convex.2012-12-21