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Instead of asking which unit-volume n-parallelotopes ($TC$, where $T\in SL_n(\mathbb{R})$, and $C=[0,1]^n$) tile $\mathbb{R}^n$ when translated by $\mathbb{Z}^n$, we can equivalently ask, which unit-d …

Let $R\subseteq P$ be the region $\{(x,y)\in P:x>0,y>0, x+y>1\}$ and let $g:P\to R$ be continuous and bijective. Let $h(p_2, p_3, p_4; (x, y)) = p_3+(p_2-p_3)y + (p_4-p_3)x$. Note that $h: P^4 \to P$ …

x is not in the interior of conv(P) iff there is an affine function $u(y) = a+ b\cdot y$ such that $u(x)\le 0$ and $u(p_i)\ge0$ for $i=0,\ldots,m$. If this problem has no feasible solution, then x is …

Like Emil noted in the comments, the question is equivalent to whether $0\in\mathrm{Conv}(\{\alpha\in\mathbb{Z}^d: \sum_{i=1}^{d}\alpha_i=0\}\setminus\{0\})$. To see that it is, note that it is the av …

No, a fixed number of added vertices can change the John ellipsoid by an arbitrary amount when added to arbitrarily many already known vertices: consider a very thin regular n-gon prism centered about …

The type of symmetry for which the simplex (not necessarily regular) is usually called "the least symmetric convex body" is the symmetry of reflection about a point (e.g. $x\mapsto-x$). There are a fe …

In the paper "Analogues of Steinitz's theorem about non-inscribable polytopes" by E. Schulte, which comes out of the collection "Intuitive Geometry" from 1987, the author seems to prove a negative res …

This is not an answer, but rather a documentation of a failed attempt to obtain an answer. One problem for which we know there is a significant difference between unrestricted sets of translations an …

No, a line connecting a vertex to the centroid is not necessarily an area bisector. This follows easily from your observation that not every line through the centroid bisects area: take a line which g …

As noted in the comments, the case where $A\subseteq\mathbb{R}^2$ is the interior of the unit square and $B$ is a side of that square satisfies the assumptions posed, but in that case neither (1) nor …

A convex body $K\subset\mathbb{R}^n$ all of whose $(n-1)$-dimensional projections have the same $n-1$-content is known as a body of constant brightness, by analogy with bodies of constant width. The t …

This follows easily from the fact that the perimeter of a convex curve is in direct proportion with the mean width, and the width of the curve is in every direction smaller than the width of the circl …

The so-called separation property of convex cones tells you that $C_1$ and $C_2$ have disjoint interiors if and only if there is a hyperplane separating them. Namely, there exists $\mathbf{y}\in \math …

There are two ways to represent convex polytopes: as the convex hull of its vertices, or as the intersection of the half-spaces whose boundaries contain the faces. If you store both of these represent …