The effective radius $er(A)$ of a $n$-solid $A$, is defined by Schramm (see the question by Gil Kalai Volumes of Sets of Constant Width in High Dimensions) to be the radius of the $n$-ball that has the same volume.
Question(s) :
Let $A$ be a $n$-convex (convex set in the $n$ dimentionnal euclidian affine space $\mathbb A_n$) and $H$ a hyperplane of $\mathbb A^n$ such that $B=A\cap H$ has maximal volume (resp. such that the boundary of $B'=A\cap H$ is maximal in volume)
a) Is $er(B)\geq er(A) $ ?
b) Is $er(B')\geq er(A)$ ?
I think the answer (especially the b) question) if positive implies the resolution of the question in Volumes of Sets of Constant Width in High Dimensions podted by Gil Kalai.
Let me try an heuristic for the first inequality, in $3$ dimensions (I write vol($A$) as well as for a volume or an area, as soon as there is no ambiguities) :
Take $t\mapsto H_t$ a path of plans parallele to $H$ that go throught $C$ and that intersect $C$ in $B_t$. Now we associate to $B_t$ a disk $D_t$ that radius is $er(B_t)$ such that its center is $t\in T$ and that $T$ is a line. Consider $E=\bigcup_{t\in T} D_t$. It is clear that $Vol (E) = Vol (C)$. It is also clear that $E$ is the $\pi$- rotation solid generated by , say $E_0$ : union of coplanar "effectives diameters"
The question a) for dim 3 is then equivalent to :
Is $vol(E_0)$ smaller then $vol(B)$ ?
This would be automatically the case if $E_0$ can be isometrically embeded in $C$ whom a greater section (in volume, well in this case "area") is $B$ - it appears that the question that translate the possibility of this transversal embedding is : can we put the line $T$ inside $C$ and take in each "efficient disk" an "efficient diametre" inside $C$ in such a way that they are co-planar.. we could imagine that there is some torsion around $T$ that make this operation impossible, despite convexity condiderations... never the less, this intermediary request is stronger then the one of the (here part a) but also wrt part b)) question...
The problem of this heuristic is not only the torsions phenomenons that I think they might not occure in 3 dim (and that we can probably get rid of, considering that the case that we are dealing with are well behaving, especially throught heredity) but the fact that I'm not sure it can be adapted to greater dimensions, because $E_0$ is 2-dimensional in any case... that does not go against the relevance of the questions but that does not give at least to me a clear hope or a clue, well maybe this discussion can be adapted to the general case... or not.
