How and whether you define concavity of a set is important in economics.

The problem with defining concavity of sets as non-convexity is that this can lead to confusion when defining functions as concave or quasi-concave, which are very important in economics. Functions which are concave or quasi-concave have nice maximums.

A function is concave if for all Geometrically, that means the set of points below the graph of is convex. So concavity (non-convexity) of a set has no relation to concavity of a function.

Similarly a function is quasi-concave if the upper level sets are convex. Again concavity (non-convexity) of a set has no relation to quasi-concavity of a function.

When teaching these definitions, the important concepts are convexity of set and concavity and quasi-concavity of functions. There is no need to bring in concavity of a set. It just leads to confusion. If you want to say a set is not convex, just say that or use the word non-convex.

What the page I referred to above did was to define concave set in a way that was useful for defining concave functions. At the time it was written (1983), this may have not been incorrect because I don't think the definition of concavity of a set was well-established then. Definitions change over time in mathematics.

Here is a quote from another page google turns up:

I used words like "convex set" and "concave set" in class. It has been pointed out that while "concave set" is a mathematical concept, it is not nearly as commonly used as "convex set", which is the common vocab including in economic applications.

There's good reason for this: In a convex set, if you join 2 points by a line segment, the line segment lies entirely within the set. In a nonconvex (concave) set, if you join 2 points, the line segment may or may not lie within the set, depending on the points you chose. This makes convexity easier to deal with.

He then goes on to discuss quasi-concave functions.