Inverse Blaschke-Santal\'o inequality for convex curves enclosing the origin several times

Abstract

H. Guggenheimer generalized the planar volume product problem for locally convex curves $C$ enclosing the origin $k \ge 2$ times. He conjectured that the minimal volume product $V(C)V(C^*)$ for these curves is attained if the curve consists of the longest diagonals of a regular $(2k+1)$-gon, with centre $0$, these diagonals taken always in the positive orientation. This conjectured minimum is of the form $k^2 + O(k)$. We investigate special cases of this conjecture. We prove it for locally convex $n$-gons with $2k+1 \le n \le 4k$, if the central angles at $0$ of all sides are equal to $2k \pi /n$. For $4k+1 \le n$ we prove that for locally convex $n$-gons enclosing the origin $k \ge 2$ times the critical (stationary) values of the volume product $V(K)V(K^*)$ are attained exactly when up to a non-singular linear map the vertices lie on the unit circle about $0$, and the central angles of all sides are equal to $2k \pi /n$. For locally convex $n$-gons enclosing the origin $k \ge 2$ times, and inscribed to the unit circle, with $2k+1 \le n$, we prove the conjecture up to a multiplicative factor about $0.43$.