Isoperimetric inequalities for closed curves in spaces of constant curvature (Q1196072)

Let \(f: C\to R_ c^ n\) denote a smooth immersion of an oriented circle \(C\) into the \(n\)-dimensional space of sectional curvature \(c\) (\(c=+1\) or \(c=-1\)), \(n\geq 3\). Let \(G\) be the homogeneous space of all unoriented \((n-2)\)-subspaces and \(dG\) its invariant density under the group of motions of \(R_ c^ n\). The winding number \(w(g)\) of \(f\) with respect to \(g\in G\), \(g\cap f(C)=\emptyset\), is the algebraic number of intersections of \(f(C)\) with either \((n-1)\)-half space bounded by \(g\). Then the author proves the isoperimetric inequality \[ L^ 2>a(c)(32\pi^ 3/O_ n O_{n-1}) \int_ G w^ 2(g)\;dG_ g, \] where \(a(c)=1/2\) for \(c=+1\) and \(a(c)=-1\) for \(c=-1\). \(O_ n\) denotes the surface area of the \(n\)-dimensional Euclidean unit sphere. Introducing the number \(n_ c(g,\bar g)\) of hitting-chords of \(f\) w.r.t. \((g,\bar g)\in G\times G\), defined as the half number of those pairs \((x,y)\in C\times C\) (\(x\neq y\)) whose corresponding chords between \(f(x)\) and \(f(y)\) intersect both \(g\) and \(\bar g\), the author obtains a more involved, though interesting, inequality.