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

Combinatorics and more 2025-01-07

Consider $\mathbb R^n$ equipped with a norm. Given a finite set of points $K$ and a point $x$ , we consider $T(x,K)$ , the sum of distances from $x$ to the points in $K$ . Next we consider the set of points $M(K)$ that attain the minimum of $T(x,K)$ . We say that the norm is nice if $M(K)$ has non empty intersection with the convex hull of $K$ .

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).