Given C, a convex closed curve that is symmetric with respect to the origin in the Euclidean plane, consider M(C), the Minkowski metric space with C as a unit circle. We are interested in the chromatic number of this space, which is the minimum number of colors required to color the points of the plane so that any two points at C-distance exactly 1 receive different colors. Chilakamarri proved that for any centrally symmetric convex curve C, the chromatic number of M(C) is between 4 and 7, and that if C is either a parallelogram or a centrally symmetric hexagon then the chromatic number of M(C) is 4. From the recent work of de Grey, it is now known that the chromatic number of the plane with the Euclidean metric is at least 5; we know however of no other curve C, for which the chromatic number of M(C) satisfies this inequality. Among other things, we will show that if C is a regular octagon then at least 5 colors are needed to properly color M(C). This is joint work with Geoffrey Exoo.