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The square case was posed as a problem at Leningrad (now St. Petersburg) high school math olympiad in 1963. I wrote a solution of this problem for the volume "St. Petersburg mathematical olympiads 196 …

$3^d-1$ and $3^d$, resp. Let $C=[0,1]^d$, Consider the $3^d$ points in $C$ all whose coordinates are from the set $\{0,\frac12,1\}$. No translated copy of $C'$ can cover two of these points, hence at …

No, even if you assume that $f$ is convex on $\mathbb R^{d+n}$. Take $n=2$ and let $y_1,\dots,y_{d+1}\in\mathbb R^2$ be vertices of a convex polygon. Let $X_1,\dots,X_{d+1}\subset\mathbb R^n$ be your …

Three regions is the minimum for any convex figure with a smooth boundary. To construct a partition, let $AB$ be the diameter of $R$. The segment $[AB]$ splits $R$ into two pieces $R_1$ and $R_2$. Let …

No, here is a counter-example (to revision 9). Let $A$ be the linear map that sends the vector $(1,1,1)$ to $V:=(100,100,100)$ and is the identity on the orthogonal complement of this vector. Then an …

Q3: Laman's theorem is the same on the sphere. Indeed, a configuration with $n$ vertices and $m$ edges is defined by a system of $m$ equations in $2n-3$ variables (there are $2n$ coordinates of point …

Here is a construction of a polygon that cannot fall through the hole. Begin with a regular $MN$-gon circumscribed around a unit circle, where $M\gg N\gg 1$ and $N$ is even. For every $M$th side, dra …

Assuming $k\le m$, the answer is $\frac{2\pi}{m}-\frac{\pi}{mk}$. I prefer to speak about red and green points on the circle rather than vectors. The angle between vectors is the (intrinsic) distance …

Let me answer Question 2. Strong version: no. Consider $[0,1]$ with distance $d(x,y)=|x-y|^{1/3}$. There is no even a triple of points with rational distances - otherwise there would be a nonzero rat …