Real three-dimensional Hlawka admissibility on [26, 32]

For every real number $p$ with $26\le p\le32$, and every three vectors $x=(x_1,x_2,x_3)$, $y=(y_1,y_2,y_3)$, and $z=(z_1,z_2,z_3)$ in $\mathbb R^3$, define $N_p(v)=\left(\sum_{i=1}^{3}|v_i|^p\right)^{1/p}$ for $v\in\mathbb R^3$ and let $C_p$ be the real supremum of the set $\left\{\dfrac{3(t^p+2)^{1/p}-3^{1/p}|2-t|}{6(t^p+2)^{1/p}-3\left(2|1-t|^p+2^p\right)^{1/p}}:\ t\in\mathbb R,\ \tfrac12\le t\le2\right\}$, where powers are real powers and the quotient is interpreted using total real division (its value is $0$ if its denominator is $0$). Then $N_p(x)+N_p(y)+N_p(z)-N_p(x+y+z)\le C_p\bigl((N_p(x)+N_p(y)-N_p(x+y))+(N_p(x)+N_p(z)-N_p(x+z))+(N_p(y)+N_p(z)-N_p(y+z))\bigr)$, with vector sums taken coordinatewise. There are no restrictions on the vectors or on equality of their $N_p$ values: the assertion includes arbitrary unequal norms, cases in which any vector or any displayed vector sum is zero, and the all-zero triple, for which the inequality is $0\le0$.