18 problems

Petty's projection conjecture. The quantity $\mathcal{F}(K)$ is minimized over all convex bodies $K$ in $\mathbb{R}^n$ if and only if $K$ is an ellipsoid.

Let $K$ be a convex body in $\mathbb{R}^n$, let $i\in\\{1,\dots,n-1\\}$, and let $\Pi_i K$ be its projection body of order $i$, defined by … where $V_i$ is the $i$th intrinsic volu…

Let $K\in\mathcal{K}_{o}^n$ be a convex body and let $p>1$. The $L^p$ projection body ${\Pi}_pK$ and the $L^p$ centroid body ${\Gamma}_pK$ are defined by their support functions, a…

Ellipsoidal minimizer conjecture. For $n\geq 3$ and $m\geq 2$, the lower bound for this quantity is attained by ellipsoids.

Brannen's conjecture. The minimum of $\Theta$ is attained if $K$ is a simplex. The paper presents this as an additional open extremal problem for the Petty functional and gives no…

Petty's conjecture. The functional $\Theta$ attains its maximum if and only if $K$ is an ellipsoid. The determination of the extreme values of $\Theta$ is presented as an open prob…

Saroglou's lower-order projection-body conjecture. For $i=1,\dots,d-2$, the functionals $P_i(\cdot)$ and $Q_i(\cdot)$ are minimized precisely for Euclidean balls. The source notes…

Let $K$ be a $d$-dimensional convex body containing the origin in its interior. Define … For $x\in S^{d-1}$, let $K\cap(sx+x^\bot)$ be the $(d-1)$-dimensional section and let…

Let $K$ be a zonoid in $\mathbb{R}^d$, let $V(K)$ be its volume, and let $\Pi^2K:=\Pi(\Pi K)$ be its second projection body. Saroglou's upper-bound conjecture. … with equality if a…

Let $P(K):=V(\Pi K)/V(K)^{d-1}$, and restrict $K$ to zonoids. A Weil body is one of the bodies described in the source as a Cartesian product of symmetric convex bodies of dimensio…

Let $P(K)=V(\Pi K)/V(K)^{d-1}$ for a convex body $K\subseteq\mathbb{R}^d$. Consider a simplex whose centroid is at the origin, and the centrally symmetric body of maximal volume co…

Makai–Martini conjecture. The minimum of $Q(K)$ is attained if and only if $K$ is a parallelepiped; in the general, not necessarily centrally symmetric case, the simplex is the ext…

For a centrally symmetric convex body $K$ in $\mathbb{R}^n$, define … and let $C_n$ be the $n$-dimensional cube. Asymptotic Schneider conjecture. The ratio … tends to $1$ as $n$ te…

Let $K$ be a convex body in $\mathbb{R}^n$, let $\Pi(K)$ denote its projection body, and define … A convex body is centrally symmetric if it is symmetric about some center. Brannen…

Let $K$ be a convex body in $\mathbb{R}^n$, let $\Pi(K)$ denote its projection body, and define the affine invariant … For centrally symmetric $K$, let $C_n$ be the $n$-dimensional…

Let $K,L\in\mathcal{K}_n$ be convex bodies, and for a unit vector $u\in\mathbb{R}^n$ let $K_u$ and $L_u$ denote their projections onto the hyperplane orthogonal to $u$. Suppose tha…

Let $K$ and $M$ be $o$-symmetric convex bodies in $\mathbb{R}^n$, let $\Pi K$ denote the projection body, and let $\delta_{\rm BM}$ denote the Banach–Mazur distance. Bourgain and L…

Lutwak–Yang–Zhang's conjecture. For every $\varphi\in{\cal C}$, every maximizer of this ratio is an ellipsoid.