2 Basic notations and facts

In this section, we recall some definitions, notations, and facts from convex geometry. A convex body K is a compact, convex subset of ${\mathbb {R}}^{d}$ with nonempty interior. A convex body K is said to be centered if it is symmetric with respect to the origin, i.e., $-K=K$ . We denote by $\mathcal K$ the set of convex bodies in ${\mathbb {R}}^d$ , and the set of centered convex bodies will be denoted by $\mathcal K_{o}$ . As usual, $S^{d-1}$ will stand for the standard Euclidean unit sphere in ${\mathbb {R}}^{d}$ . The symbol $\lambda _{i}(\cdot )$ will stand for the i-dimensional Lebesgue measure $($ volume $)$ in ${\mathbb {R}}^{d}$ , where $1\leq i\leq d$ , and when $i=d$ the subscript will be omitted. For a given direction $u\in S^{d-1}$ , we use $u^{\perp }$ to denote the $(d-1)$ -dimensional hyperplane (passing through the origin) orthogonal to u, and by $l_{u}$ the $1$ -subspace parallel to u. Furthermore, $\lambda _1 (K|l_u)$ denotes the width of K at u, and $\lambda _{d-1} (K | u^\perp )$ the $(d-1)$ -dimensional outer cross-section measure or brightness of K at $u \in S^{d-1}$ , where $K|u^\perp $ is the orthogonal projection of K onto $u^\perp $ (more about these notations, the reader should refer to [Reference Gardner5]).

For $K\in \mathcal K$ with the origin an interior point of K, its polar body $K^{\circ }$ is defined by

$$ \begin{align*}K^{\circ}=\{y\in {\mathbb{R}}^{d} : \langle x,y \rangle \leq 1, x\in K\},\end{align*} $$

where $\langle \cdot , \cdot \rangle $ denotes the standard scalar product in ${\mathbb {R}}^{d}$ .

We mention the following simple properties of $K\in \mathcal K_{o}$ : $(K^{\circ })^{\circ }=K$ , $(\alpha K)^{\circ }=(1/{\alpha })K^{\circ }$ for $\alpha>0$ . For $K_{1}\in \mathcal K_{0}$ and $K_{2}\in \mathcal K_{0}$ if $K_{1}\subseteq K_{2}$ , then $K^{\circ }_{2}\subseteq K^{\circ }_{1}$ .

The standard basis will be used to identify ${\mathbb {R}}^{d}$ and its dual space ${\mathbb {R}}^{d\ast }$ . In that case, $\lambda _{i}(\cdot )$ and $\lambda ^{\ast }_{i}(\cdot )$ coincide in ${\mathbb {R}}^d$ . The symbol $\epsilon _{i}$ will stand for the i-dimensional volume of the unit ball in ${\mathbb {R}}^{i}$ .

The support function $h_{K} : S^{d-1}\rightarrow {\mathbb {R}}$ of a convex body K is defined as $h_{K}(u)=\sup \{\langle u,y \rangle :y\in K\}$ . It is well known that $h_{K}$ is monotone with respect to inclusion (i.e., if $K\subseteq L$ , then $h_{K}\leq h_{L}$ ), and positive homogeneous (i.e., $h_{\alpha K}(u)= h_{K}(\alpha u)=\alpha h_{K}$ for all $\alpha>0$ ). Furthermore, if $0\in K$ , then $h_{K}(u)$ is the distance from the origin to the supporting hyperplane of K with outer unit normal vector u (more about support functions and properties, see [Reference Schneider16]). When the origin is an interior point of K, its radial function $\rho _{K}(u)$ is defined by $\rho _{K}(u)=\max \{\alpha \geq 0 : \alpha u\in K\}.$ The following relation between these two functions is well known:

(1) $$ \begin{align} \rho_{K^\circ}(u) = \frac{1}{h_K(u)}\,, \,\, u \in S^{d-1}\,. \end{align} $$

It is easy to observe that if K is a centered convex body, then $2\rho _{K}(u)=\lambda _{1}(K\cap l_{u})$ , and $2h_{K}(u)=\lambda _{1}(K|l_{u})$ for any $u\in S^{d-1}$ .

For $K\in \mathcal K$ , the projection body $\Pi K$ of K is defined as the convex body whose supporting hyperplane in a given direction u has a distance $\lambda _{d-1}(K|u^{\perp })$ from the origin, i.e., $h_{\Pi K}(u)=\lambda _{d-1}(K|u^{\perp })$ for each $u\in S^{d-1}$ (see [Reference Gardner5, Chapter 4]). Note that any projection body is a zonoid (i.e., a limit of vector sums of segments) centered at the origin.

The following well-known fact (see [Reference Gardner5]) will be used here for centered convex bodies; if S be a subspace of ${\mathbb {R}}^{d}$ and $B\in \mathcal K_{o}$ , then

(2) $$ \begin{align} (B\cap S)^{\circ}=B^{\circ}|S. \end{align} $$

For $B\in \mathcal K_{o}$ , the special case of Blaschke–Santaló inequality states that

$$ \begin{align*}\lambda(B)\lambda(B^{\circ})\leq \epsilon_{d}^2,\end{align*} $$

with equality if and only if B is an ellipsoid (proved by Blaschke [Reference Blaschke3] in ${\mathbb {R}}^2$ and ${\mathbb {R}}^3$ , then by Santaló [Reference Santaló15] for all dimensions; more about this inequality the reader should refer to [Reference Gardner5] and the references therein).

3 Inequalities for inscribed double cones and circumscribed cylinders

First, we give the following alternative proof of Lutwak’s minimax inequality for inscribed double cones. For the sake of simplicity, we will omit the constant $(2d)^{-1}$ in the volume equation of inscribed double cones.

Theorem 1 If $B\in \mathcal K_{o}$ , then

$$ \begin{align*}\min_{u\in S^{d-1}}\lambda_{d-1}(B\cap u^{\perp})\lambda_{1}(B|l_{u})\lambda(B^{\circ})\leq 2\epsilon_{d-1}\epsilon_{d},\end{align*} $$

with equality if and only if B is an ellipsoid.

Proof It follows from Petty’s result that there exists a unit vector $u\in S^{d-1}$ such that

(3) $$ \begin{align} \frac{\lambda_{d-1}(B^{\circ}|u^{\perp})\lambda_{1}(B^{\circ}\cap l_{u})}{\lambda(B^{\circ})}\geq\frac{2\epsilon_{d-1}}{\epsilon_{d}}. \end{align} $$

From the Blaschke–Santaló inequality and the identity (2), we have

(4) $$ \begin{align} \lambda_{d-1}(B^{\circ}|u^{\perp})\leq\frac{\epsilon_{d-1}^2}{\lambda_{d-1}((B^{\circ}|u^{\perp})^{\circ})}=\frac{\epsilon_{d-1}^2}{\lambda_{d-1}(B\cap u^{\perp})}, \end{align} $$

with equality if and only if $B^{\circ }|u^{\perp }$ is an ellipsoid.

The identity (1) can be written as $2\rho _{B^{\circ }}(u)2h_{B}(u)=4$ for every $u\in S^{d-1}$ . It is equivalent to

(5) $$ \begin{align} \lambda_{1}(B^{\circ}\cap l_{u})\lambda_{1}(B|l_{u})=4. \end{align} $$

Therefore, (3)–(5) yield that there exists $u\in S^{d-1}$ such that

$$ \begin{align*}\frac{\epsilon_{d-1}^2}{\lambda_{d-1}(B\cap u^{\perp})}\frac{4}{\lambda_{1}(B|l_{u})}\geq\lambda_{d-1}(B^{\circ}|u^{\perp})\lambda_{1}(B^{\circ}\cap l_{u})\geq\frac{2\epsilon_{d-1}}{\epsilon_{d}}\lambda(B^{\circ}).\end{align*} $$

Thus, the result follows. Obviously, if B is an ellipsoid, then equality holds. Now, assume that

$$ \begin{align*}\min_{u\in S^{d-1}}\lambda_{d-1}(B\cap u^{\perp})\lambda_{1}(B|l_{u})\lambda(B^{\circ})= 2\epsilon_{d-1}\epsilon_{d}.\end{align*} $$

Then, for every $u\in S^{d-1},$

$$ \begin{align*}\lambda_{d-1}(B\cap u^{\perp})\lambda_{1}(B|l_{u})\lambda(B^{\circ})\geq 2\epsilon_{d-1}\epsilon_{d}.\end{align*} $$

Applying (2), (4), and the Blaschke–Santaló inequality to the last inequality yields

$$ \begin{align*}\frac{\lambda_{d-1}(B^{\circ}|u^{\perp})\lambda_{1}(B^{\circ}\cap l_{u})}{\lambda(B^{\circ})}\leq\frac{2\epsilon_{d-1}}{\epsilon_{d}},\end{align*} $$

for all $u\in S^{d-1}$ . It follows from Petty’s result that $B^{\circ }$ must be an ellipsoid. Hence, B is an ellipsoid.

Problem 6 of [Reference Busemann and Petty4] asks to find a centered convex body B of ${\mathbb {R}}^d$ for which

$$ \begin{align*}\max_{u\in S^{d-1}}\frac{\lambda_{d-1}(B\cap u^{\perp})\lambda_{1}(B|l_{u})}{\lambda(B)}\end{align*} $$

is minimal. It was conjectured that the ellipsoid is the answer to this problem. Related to this conjecture, we prove the following result.

Theorem 2 Let B be a centered convex body in ${\mathbb {R}}^d$ . If

$$ \begin{align*}\max_{u\in S^{d-1}}\frac{\lambda_{d-1}(B\cap u^{\perp})\lambda_{1}(B|l_{u})}{\lambda(B)}\geq \frac{2\epsilon_{d-1}}{\epsilon_{d}},\end{align*} $$

then

$$ \begin{align*}\min_{u\in S^{d-1}}\lambda_{d-1}(B^{\circ}|u^{\perp})\lambda_{1}(B^{\circ}\cap l_{u})\lambda(B)\leq 2\epsilon_{d-1}\epsilon_{d}.\end{align*} $$

Proof It follows from the hypothesis of the theorem that there exists $u\in S^{d-1}$ such that

$$ \begin{align*}\frac{\lambda_{d-1}(B\cap u^{\perp})\lambda_{1}(B|l_{u})}{\lambda(B)}\geq \frac{2\epsilon_{d-1}}{\epsilon_{d}}.\end{align*} $$

Using the Blaschke–Santaló inequality and (2), we get

$$ \begin{align*}\lambda_{d-1}(B\cap u^{\perp})\leq\frac{\epsilon_{d-1}^2}{\lambda_{d-1}(B^{\circ}|u^{\perp})},\end{align*} $$

with equality if and only if $B\cap u^{\perp }$ is an ellipsoid. The result can be established using the identity (5). One can also see that if B is an ellipsoid, then equality holds for both, the hypothesis and the conclusion of theorem. Now, assume that

$$ \begin{align*}\min_{u\in S^{d-1}}\lambda_{d-1}(B^{\circ}|u^{\perp})\lambda_{1}(B^{\circ}\cap l_{u})\lambda(B)=2\epsilon_{d-1}\epsilon_{d}.\end{align*} $$

Then

$$ \begin{align*}\lambda_{d-1}(B^{\circ}|u^{\perp})\lambda_{1}(B^{\circ}\cap l_{u})\lambda(B)\geq 2\epsilon_{d-1}\epsilon_{d},\end{align*} $$

for all $u\in S^{d-1}$ . Applying (5), the Blaschke–Santaló inequality, and (2) to the last inequality, we get

$$ \begin{align*}\frac{\lambda_{d-1}(B\cap u^{\perp})\lambda_{1}(B|l_{u})}{\lambda(B)}\leq \frac{2\epsilon_{d-1}}{\epsilon_{d}},\end{align*} $$

for all $u\in S^{d-1}$ . Thus,

$$ \begin{align*}\max_{u\in S^{d-1}}\frac{\lambda_{d-1}(B\cap u^{\perp})\lambda_{1}(B|l_{u})}{\lambda(B)}=\frac{2\epsilon_{d-1}}{\epsilon_{d}}.\\[-42pt]\end{align*} $$

In [Reference Busemann and Petty4], it was also mentioned that the maximum value of the quantity

$$ \begin{align*}\min_{u\in S^{d-1}}\frac{\lambda_{d-1}(B\cap u^{\perp})\lambda_{1}(B|l_{u})}{\lambda(B)}\end{align*} $$

is still unsolved when $d\geq 3$ . Similar to Theorem 2, one can easily deduce the following related result.

Theorem 3 Let $B\in \mathcal K_{o}$ . If

$$ \begin{align*}\max_{u\in S^{d-1}}\lambda_{d-1}(B^{\circ}|u^{\perp})\lambda_{1}(B^{\circ}\cap l_{u})\lambda(B)\geq 2\epsilon_{d-1}\epsilon_{d},\end{align*} $$

then

$$ \begin{align*}\min_{u\in S^{d-1}}\frac{\lambda_{d-1}(B\cap u^{\perp})\lambda_{1}(B|l_{u})}{\lambda(B)}\leq \frac{2\epsilon_{d-1}}{\epsilon_{d}}.\end{align*} $$

Furthermore, if $\min _{u\in S^{d-1}}\lambda _{d-1}(B\cap u^{\perp })\lambda _{1}(B|l_{u})\lambda (B)^{-1}= 2\epsilon _{d-1}\epsilon _{d}^{-1}$ , then $\max _{u\in S^{d-1}}\lambda _{d-1}(B^{\circ }|u^{\perp })\lambda _{1}(B^{\circ }\cap l_{u})\lambda (B)=2\epsilon _{d-1}\epsilon _{d}.$

One of the challenging problems in convex geometry is that whether a centered convex body B and $\Pi B^{\circ }$ (or $B^{\circ }$ and $\Pi B$ ) are homothetic if and only if B is an ellipsoid. We mention that the isoperimetrix of Holmes-Thompson measure is defined as ${\hat {I}_B={\epsilon _{d}}\lambda ^{-1}(B^{\circ })\epsilon _{d-1}^{-1}\Pi B^{\circ }}$ (see [Reference Thompson17]). The relative inner radius $r(B, \hat {I}_B)$ of B with respect to $\hat {I}_B$ is the largest $\alpha>0$ such that $\alpha \hat {I}_B\subseteq B$ , and the relative outer radius $R(B, \hat {I}_B)$ of B with respect to $\hat {I}_B$ is the smallest $\alpha>0$ such that $B\subseteq \alpha \hat {I}_B$ . In [Reference Martini and Mustafaev11], it was shown that

(6) $$ \begin{align} R(B, \hat{I}_B)=\max_{u\in S^{d-1}}\frac{2\epsilon_{d-1}}{\epsilon_{d}}\frac{\lambda(B^{\circ})}{\lambda_{d-1}(B^{\circ}|u^{\perp})\lambda_{1}(B^{\circ}\cap l_{u})} \end{align} $$

and

(7) $$ \begin{align} r(B, \hat{I}_B)=\min_{u\in S^{d-1}}\frac{2\epsilon_{d-1}}{\epsilon_{d}}\frac{\lambda(B^{\circ})}{\lambda_{d-1}(B^{\circ}|u^{\perp})\lambda_{1}(B^{\circ}\cap l_{u})}. \end{align} $$

Theorem 4 Let B be a centered convex body in ${\mathbb {R}}^d$ . If

$$ \begin{align*}\max_{u\in S^{d-1}}\lambda_{d-1}(B\cap u^{\perp})\lambda_{1}(B|l_{u})\lambda(B^{\circ})\geq 2\epsilon_{d-1}\epsilon_{d},\end{align*} $$

then B and $\Pi B^{\circ }$ are homothetic if and only if B is an ellipsoid.

Proof Similar to Theorem 1, one can easily establish that if

$$ \begin{align*}\max_{u\in S^{d-1}}\lambda_{d-1}(B\cap u^{\perp})\lambda_{1}(B|l_{u})\lambda(B^{\circ})\geq 2\epsilon_{d-1}\epsilon_{d},\end{align*} $$

then

$$ \begin{align*}\min_{u\in S^{d-1}}\frac{\lambda_{d-1}(B^{\circ}|u^{\perp})\lambda_{1}(B^{\circ}\cap l_{u})}{\lambda(B^{\circ})}\leq\frac{2\epsilon_{d-1}}{\epsilon_{d}}.\end{align*} $$

Therefore,

$$ \begin{align*}1\leq\frac{2\epsilon_{d-1}}{\epsilon_{d}}\max_{u\in S^{d-1}}\frac{\lambda(B^{\circ})}{\lambda_{d-1}(B^{\circ}|u^{\perp})\lambda_{1}(B^{\circ}\cap l_{u})}.\end{align*} $$

Hence, the identity (6) yields $R(B, \hat {I}_{B})\geq 1.$

Applying Petty’s result to (7), we obtain that $r(B, \hat {I}_{B})\leq 1$ , with equality if and only if B is an ellipsoid (see also [Reference Thompson17]). One can observe that B and $\Pi B^{\circ }$ are homothetic if and only if $r(B,\hat {I}_{B})=R(B, \hat {I}_{B})$ . It could be the case only if B is an ellipsoid.