We consider the local maximizer $x^TAx$ over simplex.
$A$ is a symmetric $N \times N$ matrix with 4 blocks.
- First block $L \times L$ with elements $0 <a_{ij}<0.5$
- Second block $K \times K$ with elements $0 <a_{ij}<0.5$
- Third block $L \times K$ with elements $0.5<a_{ij}<1$
- Fourth block $K \times L$ with elements $0.5<a_{ij}<1$
except for diagonal elements that are zero, $a_{ii}=0$ for $1\leq i \leq N$. Note $L+K=N$.
Can we prove that there is a gap, meaning a margin or difference, between the first $L$ elements of $x_i$: for $1\leq i \leq L$ AND the rest of elements ($K$ elements) $x_i$ for $L< i \leq L+K=N$. Can we quantify this gap?