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

Solution 1:

The inequality that I know under the name isodiametric inequality is $$ \frac{\text{vol}(K)}{\text{diam}(K)^d} \le \frac{\text{vol}(B)}{\text{diam}(B)^d} $$ for any convex body $K$ in $\mathbb{R}^d$, where $B$ denotes the unit ball.

Proof 1: By Steiner symmetrization (which preserves volume, decreases diameter, and tends to the ball if desired). Proof 2: If $K$ has diameter at most 2 then $K-K\subseteq 2B$; by the Brunn-Minkowski inequality, $\text{vol}(K)\le\text{vol}(\frac12(K-K))$; thus $\text{diam}(K)\le\text{diam}(B)\implies \text{vol}(K)\le\text{vol}(B)$, which is equivalent to the desired inequality. Note that proof 2 doesn't really use the fact that $B$ is the Euclidean ball: it actually proves the more general analogous statement where we take $B$ to be any origin-symmetric convex body and measure diameters in the norm whose unit ball is $B$. (Proof 2 also yields that this isodiametric inequality is actually equivalent to the special case of Brunn-Minkowski that was used.) All of the above is in Gruber's recent book on convex geometry, for example.

Another proof of the generalization to arbitrary norms was given by M. S. Mel'nikov ("Dependence of volume and diameter of sets in an $n$-dimensional Banach space", Uspekhi Mat. Nauk 18(4) 165–170, 1963, http://mi.mathnet.ru/eng/umn6384): the key fact in that proof is that if the diameter of $K$ (in the sense of $B$) is at most 2 then the diameter of $K_t$ (in the sense of $B_t$) is also at most 2, where $K_t$ denotes the level set of height $t$ of the projection of $K$ (as a density) onto a fixed hyperplane; this allows a proof by induction on the dimension, and it anticipates the proof of the Prékopa-Leindler inequality, a generalization of Brunn-Minkowski. (For Prékopa-Leindler, see lecture 5 in Keith Ball's An Elementary Introduction to Modern Convex Geometry.)

Another inequality of the type you've asked about is Urysohn's inequality: $$ \frac{\text{vol}(K)}{w(K)^d} \le \frac{\text{vol}(B)}{w(B)^d} $$ for any convex body $K$ in $\mathbb{R}^d$, where $B$ denotes the Euclidean unit ball and $w(\cdot)$ denotes mean width. (This time it really matters that it's the Euclidean ball.) Since $w(K)\le\text{diam}(K)$, this is a strengthening of the isodiametric inequality above.

Proof 1: Steiner symmetrization reduces mean width. Indeed, if $S_u$ denotes Steiner symmetrization wrt the hyperplane orthogonal to a unit vector $u$, and $R_u$ denotes reflection in that hyperplane, then $h_{S_u(K)}(\theta)=\frac12 h_K(\theta)+\frac12 h_K(R_u(\theta))$, where $h_K$ denotes the support functional of $K$; now integrate over $\theta\in S^{d-1}$ and use Jensen's inequality. (I got this from some unpublished notes of Giannopoulos.) Proof 2: See Pisier's book The Volume of Convex Bodies and Banach Space Geometry (Cambridge UP, 1989, p.6; Pisier writes that he learned this proof from Vitali Milman). In short, you generalize Minkowski addition of sets to Minkowski integration of set-valued functions, and you get an analogue of Brunn-Minkowski: $$ \int_\Omega \text{vol}(A_t)^{1/n} \,d\mu(t) \le \text{vol}\left(\int_\Omega A_t \,d\mu(t)\right)^{1/n} $$ when $\mu$ is a probability measure and everything is suitably measurable. By symmetry, $\int_{O(d)} TK \,d\mu(T)$ is some multiple of the Euclidean ball (here $O(d)$ is the orthogonal group on $\mathbb{R}^d$, and $\mu$ is its Haar probability measure); a computation shows it's actually $\frac12 w(K)B$, and the Brunn-Minkowski analogue above finishes the proof.

As requested in comments, here's a generalization to other intrinsic volumes: $$ 1\le i\le j\le d\implies \frac{V_i(B)^{1/i}}{V_j(B)^{1/j}} \le \frac{V_i(K)^{1/i}}{V_j(K)^{1/j}}$$ (The case $i=1$, $j=d$ is Urysohn's inequality.) Proof: A special case of the Alexandrov-Fenchel inequality is $$ W_i(K)^2 \ge W_{i-1}(K) W_{i+1}(K) \tag{$\ast$} $$ where $W_i(\cdot)$ denotes quermassintegrals: $$ W_i(K) = V(\underbrace{K,\dotsc,K}_{d-i},\underbrace{B,\dotsc,B}_i) = \frac{\kappa_i}{\binom di} V_{d-i}(K) $$ where $\kappa_i$ is the volume of the $i$-dimensional unit Euclidean ball. It follows that $$ i\mapsto\left(\frac{W_d(K)}{W_{d-i}(K)}\right)^{1/i} \tag{$\dagger$} $$ is an increasing function for $1\le i\le d$. (You can just prove the $i$-vs-$(i+1)$ case by induction on $i$, but what's really going on here is that $i\mapsto\log W_i(K)$ is "concave" — scare quotes because its domain is discrete. The inequality ($\ast$) is the local version of this, analogous to saying that the second derivative is nonpositive; that ($\dagger$) is increasing means that the slopes over $[d-i,d]$ are increasing with $i$.) But $W_d(K) = \text{vol}(B) = W_{d-i}(B)$, so a bit of rearrangement yields the desired inequality.

(Unfortunately I'm not familiar with the literature around Alexandrov-Fenchel, so I can't give good references here.)

You might also want to consider things like the reverse isoperimetric inequality, which asserts that (1) every centrally symmetric convex body $K$ has an affine image $K'$ such that $$ \frac{V_d(K')^{1/d}}{V_{d-1}(K')^{1/(d-1)}} \ge \frac{V_d(B_\infty^d)^{1/d}}{V_{d-1}(B_\infty^d)^{1/(d-1)}} $$ where $B_\infty^d$ is the cube $[-1,1]^d$ (i.e., the unit ball of the $\ell_\infty^d$ norm), and that (2) every convex body $K$ has an affine image $K'$ such that $$ \frac{V_d(K')^{1/d}}{V_{d-1}(K')^{1/(d-1)}} \ge \frac{V_d(\Delta)^{1/d}}{V_{d-1}(\Delta)^{1/(d-1)}} $$ These inequalities are due to Keith Ball (see lecture 6 of his book mentioned above for the proof and references), relying on John's theorem and a normalized version of the Brascamp-Lieb inequality. For a proof of Brascamp-Lieb in the needed form (and much more besides, including equality cases in the above reverse isoperimetric inequalities), see F. Barthe, "On a reverse form of the Brascamp-Lieb inequality", arxiv:math/9705210. (A simplified version for the needed one-dimensional special case appears in K. Ball, "Convex geometry and functional analysis", in volume 1 of Handbook of the Geometry of Banach Spaces, Johnson and Lindenstrauss (eds.), North-Holland, 2001.)

Solution 2:

I think my question is already really long so I add here the other inequalities I find out.

• In the question relation between mean-width and diameter it is pointed out that that for any convex body $K$ in $\mathbb{R}^d$ with positive diameter we have

$$\frac{1}{\sqrt{2\pi d}} \simeq\frac{\textrm{Mean-width}(L)}{\textrm{Diameter}(L)} \leq \frac{\textrm{Mean-width}(K)}{\textrm{Diameter}(K)} \leq \frac{\textrm{Mean-width}(\textrm{Ball})}{\textrm{Diameter}(\textrm{Ball})} =1 $$

where $L$ is any line segment. The lower bound is clear if you pick $L$ to be a segment realizing the diameter of $K$. The upper bound follows from the observation that the diameter is the same as the maximal width.

• Let $\Sigma$ be a size functional of degree $k$, for any convex body $K$ with out-radius positive we have

$$\frac{\Sigma(K)^{1/k}}{\textrm{Out-radius}(K)} \leq \frac{\Sigma(\textrm{Ball})^{1/k}}{\textrm{Out-radius}(\textrm{Ball})} =\Sigma(\textrm{Unit-Ball})^{1/k}$$

where $\textrm{Ball}$ is any ball and $\textrm{Unit-Ball}$ any ball of radius $1$. This is easily seen if we consider the smallest ball containing $K$, it has the same out-radius as $K$ and its $\Sigma$-measure is bigger because $\Sigma$ is increasing under set inclusion.

• For really similar reasons, if $\Sigma$ is a size functional of degree $k$ we have for any convex body $K$ with positive in-radius

$$\Sigma(\textrm{Unit-Ball})^{1/k} =\frac{\Sigma(\textrm{Ball})^{1/k}}{\textrm{In-radius}(\textrm{Ball})} \leq \frac{\Sigma(K)^{1/k}}{\textrm{In-radius}(K)}.$$

• The Blaschke–Santaló inequality can be viewed as an inequality of this kind when we restrict the set of convex bodies to those which are centrally symmetric. For simplicity we will also assume that the center of symmetry is the origin. So we consider $K$ a convex body such that $K=-K$. We note $K^\circ:=\left\{ x\mid x\cdot y\leq 1 \text{ for all } y\in B \right\}$ its polar body. The Mahler volume of $K$ is the product of the volumes of $K$ and its polar body, namely: $V(K) V(K^\circ)$. This is invariant under linear isomorphism. The Blaschke–Santaló inequality states that the centrally symmetric shapes with maximum Mahler volume are the spheres and ellipsoids. But if we observe that $K\to V(K^\circ)^{-1}$ is a size functional of degree $d$ (we need the inverse in order to make it increasing under set inclusion). The inequality can be written

$$\frac{V(\mathrm{K})^{1/d}}{\left(V(\mathrm{K}^\circ)^{-1}\right)^{1/d}} \leq \frac{V(\mathrm{Ball})^{1/d}}{\left(V(\mathrm{Ball}^\circ)^{-1}\right)^{1/d}}$$ for all centrally symetric convex body $K$ with positive volume.

• On the other hand and with the same assumptions as in the last paragraph, the shapes with the minimum known Mahler volume are hypercubes, cross polytopes, and more generally the Hanner polytopes which include these two types of shapes, as well as their affine transformations. The Mahler conjecture states that the Mahler volume of these shapes is the smallest of any n-dimensional symmetric convex body; it remains unsolved (the whole last sentence is a direct citation from wikipedia). Again this conjecture can be written with the point of view of the question:

$$\frac{V(\mathrm{Hypercube})^{1/d}}{\left(V(\mathrm{Hypercube}^\circ)^{-1}\right)^{1/d}}\leq \frac{V(\mathrm{K})^{1/d}}{\left(V(\mathrm{K}^\circ)^{-1}\right)^{1/d}}.$$ for all centrally symetric convex body $K$ with positive volume.

Solution 3:

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.