AlphaEvolve problems·alphaevolve:9

Kakeya needle problem

Let $n \\geq 2$. Let $C^T(n)$ denote the minimal area $|\\bigcup_{j=1}^n T_j|$ of a union of triangles $T_j$ with vertices $(x_j,0)$, $(x_j + 1/n, 0)$, $(x_j + j/n, 1)$ for some real numbers $x_1,\\dots,x_n$, and similarly define $C^P(n)$ denote the minimal area $|\\bigcup_{j=1}^n P_j|$ of a union of parallelograms $P_j$ with vertices $(x_j,0), (x_j+1/n,0), (x_j+j/n,1), (x_j+(j+1)/n,0)$ for some real numbers $x_1,\\dots,x_n$. Finally, define $S^T(n)$ to be the maximal "score" $$ \\frac{\\sum_{i=1}^n |T_i|}{\\left(\\sum_{i=1}^n \\sum_{j=1}^n |T_i \\cap T_j|\\right)^{1/2} |\\bigcup_{i=1}^n T_i|^{1/2}}$$ over triangles $T_i$ as above, and define $S^P(n)$ similarly. Establish upper and lower bounds for $C^T(n)$, $C^P(n)$, $S^T(n)$, $S^P(n)$ that are as strong as possible.

Source

Something wrong or missing here? Request a correction or add a claim, with its sources.