# Isoperimetric inequality, isodiametric inequality, hyperplane conjecture… what are the inequalities of this kind known or conjectured?

I duplicate here a question I asked on math.stackexchange.

Question: Which inequalities similar to the famous isoperimetric inequality is known? conjectured?

I recently learned about some inequalities which are all similar to the famous isoperimetric inequality. Each time we consider two size functionals $\Sigma$ and $\Sigma'$ and along all the convex bodies (convex and compact) $K$ in $\mathbb{R}^d$ satisfying $\Sigma'(K)=1$, we give a bound for $\Sigma(K)$. For example in $\mathbb{R}^2$, with $\Sigma=\mathrm{Area}$ and $\Sigma'=\mathrm{Perimeter}$ we have an upper-bound given by the famous isoperimetric inequality.

If $\Sigma$ (resp. $\Sigma'$) is homogeneous of degree $k$ (resp. $k'$). The problem is equivalent to giving a bound to $$\frac{\Sigma(K)^{1/k}}{\Sigma'(K)^{1/k'}}$$ for all $K$ with $\Sigma'(K)\neq 0$. Below I list the inequalities I encountered and give a quite general definition of what I consider size functionals.

where $V_d$ is the $d$-dimensional volume, $V_{d-1}$ the $(d-1)$-intrisic volume (twice the perimeter if $d=2$ and twice the surface area if $d=3$), and $\mathrm{Ball}$ is any $d$-dimensional ball.

• The isodiametric inequality state that for any convex body $K$ in $\mathbb{R}^2$ with positive perimeter we have

$$\frac{\mathrm{Diameter}(\mathrm{Disk})}{\mathrm{Perimeter}(\mathrm{Disk})} \leq\frac{\mathrm{Diameter}(K)}{\mathrm{Perimeter}(K)} \leq\frac12$$

where $\mathrm{Diameter}(K)$ is the maximum distance between two points of $K$. 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. I guess this inequality is true in higher dimensions but I have no reference.

• Jung's theorem states that for any convex body $K$ in $\mathbb{R}^d$ with positive diameter we have the second of the following inequalities (the first is obvious)

$$\frac{\mathrm{Outradius}(\mathrm{Disk})}{\mathrm{Diameter(\mathrm{Disk})}}\leq \frac{\mathrm{Outradius}(K)}{\mathrm{Diameter(K)}}\leq \frac{\mathrm{Outradius}(\Delta_d)}{\mathrm{Diameter(\Delta_d)}}$$

where $\Delta_d$ is the $d$-dimensional regular simplex.

• The hyperplane conjecture states there exists a universal constant $C$ such that in any dimension, for any convex body $K$ in $\mathbb{R}^d$ with positive volume, we have

$$C\leq\frac{\mathrm{MaxSection}(K)^{1/(d-1)}}{\mathrm{Volume(K)}^{1/d}}<\infty$$

where $\mathrm{MaxSection}(K)=\max\left(V_{d-1}(K\cap H) : H \text{ any hyperplane of }\mathbb{R}^d\right)$ is the maximal hyperplane section of $K$.

More generally if we note $\mathcal{K}=\mathcal{K}_d$ the set of convex body of $\mathbb{R}^d$ we can consider any size functional $\Sigma:\mathcal{K}\to\mathbb{R}_{\geq 0}$ satisfying the following natural axioms:

1. $\Sigma$ is continuous,
2. not identically zero,
3. homogeneous of some degree $k$, that is: $\Sigma(\lambda K)=\lambda^k \Sigma(K)$.
4. increasing under set inclusion, that is: $(K\subset M \Rightarrow \Sigma(K)\leq\Sigma(M)$
5. invariant under translation, that is: $\Sigma(K+x)=\Sigma(K)$.

This covers most of the size functionals we usually consider:

• volume = area in dimension 2,
• surface area =perimeter in dimension 2,
• mean-width, min-width, max-width (=diameter),
• width with a given direction
• in-radius : the radius of the biggest ball include in $K$,
• out-radius : the radius of the smalles ball include in $K$,
• intrinsic volumes
• the maximal hyperplane section: $\max\left(V_{d-1}(K\cap H) : H \text{ any hyperplane of }\mathbb{R}^d\right)$
• ...

Now for any choice of couple of size functionals $\Sigma$ and $\Sigma'$ of degree $k$ and $k'$, if $K$ is a convex body with $\Sigma'(K)\neq0$ the fraction $$\frac{\Sigma(K)^{1/k}}{\Sigma'(K)^{1/k'}}\in[0,\infty[$$ is invariant under translation or rescaling of $K$.

I am interested by lower or upper bound for such fraction once we have fixed the dimension $d$ and $\Sigma$ and $\Sigma'$.

• More possibilities for $\Sigma$: isotropic constant, packing density (a/k/a critical determinant if restricting to lattice packing), covering density. – Yoav Kallus Apr 14 '14 at 14:56
• @YoavKallus I actually don't know when exactly we should make a community wiki but what you say makes sense. So I would not be surprise that the moderators turn it into a community wiki. – Gilles Bonnet Apr 14 '14 at 15:34
• @YoavKallus thanks for the possibilities you suggested to add for $\Sigma$, unfortunately I don't know the definitions of these (sorry for my ignorance). – Gilles Bonnet Apr 14 '14 at 15:39