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.
- 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 $\Sigma'$.