The Injectivity Radius of Souls of Alexandrov Spaces

Abstract A sharp lower bound for the injectivity radius in noncompact nonnegatively curved Riemannian manifolds involving their soul goes back to Šarafutdinov. We generalize this bound to the setting of Alexandrov spaces. Our main theorem reads as follows. If the injectivity radius of an Alexandrov space of nonnegative curvature does not coincide with the one of its souls, then it is at least $$ \pi K^{-1/2} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>π</mml:mi> <mml:msup> <mml:mi>K</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , where K is an upper curvature bound. We introduce the soul of Alexandrov spaces in some detail and compare two notions of injectivity radii.