# Finding the minimum sum

All numbers considered are positive real.

Given:

M spaces exist

In each space, N sets of the form (a,b,c) exist.

Now we have the following equations: $({a_{11}} + {b_{11}} + {c_{11}}) + ({a_{21}} + {b_{21}} + {c_{21}}) + ... + ({a_{N1}} + {b_{N1}} + {c_{N1}}) = {S_1}$ for space 1 … … $({a_{1M}} + {b_{1M}} + {c_{1M}}) + ({a_{2M}} + {b_{2M}} + {c_{2M}}) + ... + ({a_{NM}} + {b_{NM}} + {c_{NM}}) = {S_M}$ for space M

We know that the sum $S_Y$ is minimum. Then can we say that if we deduct a constant x from each middle element of each set of each space, then for the same combination of sets, we will get the minimum value? I.e. the new $S_{Y}$ will remain the minimum sum?

The new equations after deduction are: $({a_{11}} + {b_{11}} - x + {c_{11}}) + ({a_{21}} + {b_{21}} - x + {c_{21}}) + ... + ({a_{N1}} + {b_{N1}} - x + {c_{N1}}) = {S_1}$ … … $({a_{1M}} + {b_{1M}} - x + {c_{1M}}) + ({a_{2M}} + {b_{2M}} - x + {c_{2M}}) + ... + ({a_{NM}} + {b_{NM}} - x + {c_{NM}}) = {S_M}$

Note by Santanu Banerjee
1 month, 2 weeks ago

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I suppose the answer is obviously yes. Remove the $x$ from the brackets, and you would get $nx$. Since $S_Y\ge S_{\text{random}}$, $S_Y-nx\ge S_{\text{random}}-nx$ wouldn’t it? :)

- 1 month, 2 weeks ago