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
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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
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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.
Best Answer
For the matrix [1 2 4] , $(a=1,b=1)$, you must choose third column to satisfy the first condition but then second condition isn't satisfied. So you can choose $0$ rows which satisfy both conditions but that is less than $k\cdot a$ for any $k>0$
So answer is no.