The volume of a slightly curved submanifold in a convex region
The volume of a slightly curved submanifold in a convex region
Let <italic>T</italic> be a compact convex region in an <italic>n</italic>-dimensional Riemannian space, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k Subscript s"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>k</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{k_s}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the minimum sectional curvature in <italic>T</italic>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>κ<!-- …