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Let $K,L\subset\mathbb R^d$ be two disjoint compact convex sets with non-empty interiors. Can $x=0$ be a point of local minimum for the function $F(x)=\text{vol}_d(\text{conv(K,L+x))}$? I was asked t …

When thinking of some other problem, I stumbled upon the following innocently looking question that is natural enough to have been considered (and, possibly, solved) many years ago. However my attempt …

In this remarkable paper 30 pages are occupied by the proof of the following innocently looking lemma: Let $K$ be an origin-symmetric convex body in $\mathbb R^3$. There exist three planes through th …

In 1970 or so, Klee asked if a convex body in $\mathbb R^n$ ($n\ge 3$) whose maximal sections by hyperplanes in all directions have the same volume must be a ball. The counterexample in $\mathbb R^4$ …

Let $K$ be a bounded symmetric ($-K=K$) open convex body in $\mathbb R^n$. The critical determinant $d(K)$ of $K$ is the least possible volume $|\operatorname{det}(a_1\dots a_n)|$ of the fundamental p …