English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

H. Guggenheimer generalized the planar volume product problem for locallyconvex curves $C$ enclosing the origin $k \ge 2$ times. He conjectured that theminimal volume product $V(C)V(C^*)$ for these curves is attained if the curveconsists of the longest diagonals of a regular $(2k+1)$-gon, with centre $0$,these diagonals taken always in the positive orientation. This conjecturedminimum is of the form $k^2 + O(k)$. We investigate special cases of thisconjecture. 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 $V(K)V(K^*)$ areattained exactly when up to a non-singular linear map the vertices lie on theunit 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, andinscribed to the unit circle, with $2k+1 \le n$, we prove the conjecture up toa multiplicative factor about $0.43$.

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