Test Your Intuition (57): Are All Norms Nice?

Consider equipped with a norm. Given a finite set of points and a point , we consider , the sum of distances from to the points in . Next we consider the set of points that attain the minimum of . We say that the norm is nice if has non empty intersection with the convex hull of .

Test your intuition (57): Are there norms which are not nice?

(Added later:) Test your intuition (57)b: Which norms are nice and which norms are not nice?

Here is an illustrative representation of “nice norms,” combining societal harmony with mathematical hints (ChatGPT).