Maximum Neighbouring Sums

The vertices of a regular 10-gon are labeled $V_1, V_2, \ldots V_n$, which is a permutation of $\{ 1, 2, \ldots, 10\}$. Define a neighboring sum to be the sum of 3 consecutive vertices $V_i, V_{i+1}$ and $V_{i+2}$ [where $V_{11}=V_1, V_{12}=V_2$]. For each permutation $\sigma$, let $N_\sigma$ denote the maximum neighboring sum. As $\sigma$ ranges over all permutations, what is the minimum value of $N_\sigma$?

Details and assumptions

If the integers are written as $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$ around the circle, then the neighboring sums are $6, 9, 12, 15,$ $18, 21, 24,$ $27, 20, 13$, and the maximum neighboring sum is 27.