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I have recently heard about the hyperplane conjecture and I would like to understand better the problematic behind this conjecture.

The hyperplane conjecture: There exists a universal constant $C>0$ such that
$$Cleq C_d:=infBig( max big(V_{d-1}(Hcap K) : Hinmathcal{H}_{d}big) : K text{ convex in }mathbb{R}^d text{ and } V_d(K)=1Big )quad forall dgeq2$$
where $mathcal{H_{d}}$ is the set of hyperplane in $mathbb{R}^d$ and $V_d$ is the $d$-dimensional volume.

Basically it says that the maximal hyperplane section of any convex body cannot be really small compare to its volume. For me, it looks similar to an isoperimetric inequality.

I have several questions and instead of opening many different questions I post them all here hoping to get answer for at least some of them.
Partial answers and comments related to the conjecture but not to one of the following questions are really welcome.

1. Is there a simple justification why we should have $C_d>0$ for any $d$?


2. Which convex body (of volume $1$) has the smallest maximal section in dimension $2$? (I guess the disk, but maybe I have wrong intuition).


3. Which convex body (of volume $1$) has the smallest maximal section in dimension $3$?


4. What is the maximal hyperplane section of a unit cube?


5. 
   

   Can you give other examples of maximal sections of some specific convex body in low dimension.

   
   
   

   5 bis:Can you give an example of convex body (of volume $1$) with a maximal hyperplane section smaller then the one of the ball (of volume $1$).

   
   


6. Where does this conjecture comes from?


7. Apart from giving a lower bound, do we know anything about the behaviour of the sequence $(C_d)_{dgeq2}$? Is it decreasing? Why?


8. Is there any survey about this conjecture?

Answers:

1. ..


2. 
   Emanuele Paolini explained why it is the isodiametric problem. The disc is the solution of this problem. It has been proved by Bieberbach in 1915 (in german), I found this reference in the introduction of the article Isodiametric Problems for Polygons by by Michael J. Mossinghoff.


3. ..


4. Theorem 6 of "Volumes of sections of cubes and related problems" of Keith Ball states the maximal section of a unit cube in $mathbb{R}^d$ is $sqrt{2}$.
   This sections is the generalization of the diagonal of a square. More precisely, it is spanned by a $(d-2)$-dimensional face of the cube and the diagonal of a $2$-dimensional orthogonal face.


5. 
   

   Einar Rødland gives the maximal section for the clear example of the sphere of dimension $d$ and volume $1$: $frac{Gamma(d/2+1)^{frac{d-1}{d}}}{Gamma(d/2+1/2)} rightarrow sqrt{e}$ (when $dtoinfty$).
   Other less clear examples are really welcomed to complete the list.

   
   
   

   5 bis: When the dimension is high enough ($dgeq 10$), the cube has a smaller maximal hyperplane section then the ball because $sqrt{2}<sqrt{e}$.

   
   

About 2. In the plane the $1$-volume of the maximal hyperplane section, is nothing else than the diameter of the set. In fact since the set is convex, every section is a segment, and every diameter is a section. So the problem becomes the isodiametric problem, whose solution is known to be the disk.

My gut feeling is that the shape that would give the smallest hyperplane section would be the $d$-ball. If that is the case, the $d$-volume of a $d$-ball of radius $r$ is $nu_dr^d$ where $nu_d=pi^{d/2}/Gamma(d/2+1)$. If $r_d=nu_d^{-1/d}$ is the radius that gives volume 1 and we plug this into the $d-1$-volume of the equatorial hyperplane section $nu_{d-1}r_d^{d-1}$, we get the ratio
$$
C_dstackrel{?}{=}nu_{d-1}r_d^{d-1}
=frac{Gamma(d/2+1)^{frac{d-1}{d}}}{Gamma(d/2+1/2)}
rightarrow e^{1/2} text{ as }drightarrowinfty
$$
unless I somehow messed up the calculations here.

The logic behind this gut feeling is that any perturbation of the $d$-ball which preserves the $d$-volume would have to increase the $d-1$-volume of at least some of the sections.

If this is true, it would make $C_d>0$ and $C>0$ with good margin.

Of course, I may be wrong and the hypothesis still be true...