Roundness properties of ultrametric spaces
  We obtain several new characterizations of ultrametric spaces in terms of roundness, generalized roundness, strict p-negative type, and p-polygonal equalities (p > 0). This allows new insight into the isometric embedding of ultrametric spaces into Euclidean spaces. We also consider roundness properties additive metric spaces which are not ultrametric. 
Timothy Faver, Katelynn Kochalski, Mathav Murugan, Heidi Verheggen,   Elizabeth Wesson, and Anthony Weston
math.FA math.GN