Uniqueness of the optimizer in a general convex set

Determine whether the maximizer of the functional C↦L(C)^3 Area(C)^λ over compact convex subsets C of a given compact convex planar set K is unique for every K and every λ>0.

Background

The conjectured probability asymptotics and limit-shape statement are formulated under the condition that the maximizer set argmax PLK consists of a single element.

The paper establishes nonemptiness and several structural properties of the maximizer set, but does not establish uniqueness in general. The authors explicitly note that uniqueness is clear for some convex sets but remains uncertain for arbitrary K.

References

but to get this conclusion we would need to use also that $\argmax \PLK$ is reduced to a single element, which is clearly true for some $\K$, but we are not totally convinced that this is true for all $\K$.

— Conditioning random points by the number of vertices of their convex hull: the bi-pointed case  (2510.26330 - Marckert et al., 30 Oct 2025) in Footnote following the proposed general-case asymptotic, Section 'Construction of the conjectures in the case $\QK_{n,\floor{n\lambda}}$'