page_title: CAT(k) space
section_title: Alexandrov's comparison inequality
hierarchical_section_title: CAT(k) space / Surfaces of non-positive curvature / Alexandrov's comparison inequality
context_page_description: In mathematics, a space, where is a real number, is a specific type of metric space. Intuitively, triangles in a space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature. In a space, the curvature is bounded from above by. A notable special case is; complete spaces are known as "Hadamard spaces" after the French mathematician Jacques Hadamard.
  Originally, Aleksandrov called these spaces “ domain”.
  The terminology was coined by Mikhail Gromov in 1987 and is an acronym for Élie Cartan, Aleksandr Danilovich Aleksandrov and Victor Andreevich Toponogov.
context_section_description: The simplest form of the comparison inequality, first proved for surfaces by Alexandrov around 1940, states that
  The distance between a vertex of a geodesic triangle and the midpoint of the opposite side is always less than the corresponding distance in the comparison triangle in the plane with the same side-lengths.
  The inequality follows from the fact that if c(t) describes a geodesic parametrized by arclength and a is a fixed point, then
  f(t) = d(a,c(t))² − t²
  is a convex function, i.e.
  Taking geodesic polar coordinates with origin at a so that ‖c(t)‖ = r(t), convexity is equivalent to
  Changing to normal coordinates u, v at c(t), this inequality becomes
  u² + H⁻¹Hᵣv² ≥ 1,
  where (u,v) corresponds to the unit vector ċ(t). This follows from the inequality Hᵣ ≥ H, a consequence of the non-negativity of the derivative of the Wronskian of H and r from Sturm–Liouville theory.
caption: The median in the comparison triangle is always longer than the actual median.