In a matrix with $a$ rows and $2b+1$ columns, each cell contains a nonnegative real number. Does there exist a fixed $k>0$, independent of $a$ and $b$, for which we can always choose a set of columns $C$, with $|C|=c\in [1,2b]$, so that
For at least $k\cdot a$ rows, the sum of the $c$ numbers in each row is at least the sum of any other $2b-c$ (out of the remaining $2b+1-c$) numbers in the same row
For at least $k\cdot a$ rows, the sum of the remaining $2b+1-c$ numbers in each row is at least the sum of any other $c-1$ (out of the original $c$) numbers in the same row?
If we have only the first condition, an averaging argument using $c=b$ gives us that $k\geq 1/4$. But with the second condition also imposed, it doesn't look like the averaging argument still works.