Looking for the min-max

To minimize the maximum $m$ of five expressions, we try to make all of them equal. Thus $m = a+b+c = b+c+d = c+d+e = d+e+f = e+f+g.$ From this it follows that $a = d = g$ , $b =e$ , and $c = f$ .

We are told that the sum $a + \cdots + g = 1$ ; expression everything in $a, b, c$ we get $3a + 2b + 2c = 1,$ or $3m = 1 + b + c.$ To minimize $m$ we must therefore minimize $b+c$ , which obviously yields $b = c = 0$ . We find $m = a = \frac 13 \approx \boxed{0.333}.$