5 problems

Let $K\subset\mathbb{R}^n$ be an $o$-symmetric convex body, and let $\Pi K$ be its projection body, defined by … Its polar projection body is $\Pi^\circ K=(\Pi K)^\circ$. Petty's c…

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 $K,L\in\mathcal{K}_n$ 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…