TITLE: Isoperimetric inequality, isodiametric inequality, hyperplane conjecture... what are the inequalities of this kind known or conjectured?
QUESTION [16 upvotes]: 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.

The classical isoperimetric inequality in higher dimensions states that for any convex body $K$ in $\mathbb{R}^d$ with positive $(d-1)$-intrinsic volume we have


$$0<\frac{V_d(K)^{1/d}}{V_{d-1}(K)^{1/(d-1)}}\leq \frac{V_d(\mathrm{Ball})^{1/d}}{V_{d-1}(\mathrm{Ball})^{1/(d-1)}}$$


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:

$\Sigma$ is continuous,
not identically zero,
homogeneous of some degree $k$, that is: $\Sigma(\lambda K)=\lambda^k \Sigma(K)$.
increasing under set inclusion, that is: $K\subset M \Rightarrow \Sigma(K)\leq\Sigma(M)$
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 $\Sigma'$.

REPLY [4 votes]: Another inequality very similar to the isoperimetric inequality is the systolic inequality in its various forms. This can be restated as an inequality for convex bodies in certain cases, such as a centrally symmetric body in $R^3$. 
The earliest published inequality of this type of Pu's inequality for a real projective plane with an artbitrary Riemannian metric, which asserts that $$L^2\leq\frac{\pi}{2}A$$ where $A$ is the total area and $L$ is the least length of a noncontractible loop on the real projective plane. 
If the metric has positive Gaussian curvature it can be realized as the antipodal quotient of a convex surface in $R^3$ and then $L$ can be characterized in terms of the least distance from a point to its antipodal. There are various generalisations and a recent monograph devoted to this subject.