A rough exponent bound $K_p\le p$ for the cyclic candidate constant

For a real exponent $p$ and $t\ge0$, let $A_p(t)=(t^p+2)^{1/p}$, $B_p(t)=(2|1-t|^p+2^p)^{1/p}$, and

be the cyclic ratio built from the cyclic vectors $(-t,1,1),(1,-t,1),(1,1,-t)$ and their pairwise sums. Let

be the cyclic candidate constant. Here, for a finite index set $\iota$ and $x:\iota\to\mathbb R$, $\|x\|_p=\bigl(\sum_i|x_i|^p\bigr)^{1/p}$ is the coordinate $p$-norm, and for $t\ge0$, $A_p(t)$ is the value of $\|\cdot\|_p$ on each cyclic vector and $B_p(t)$ its value on each of their pairwise sums. For $x,y,z:\iota\to\mathbb R$ let $\mathrm{tripleGap}(x,y,z)=\|x\|_p+\|y\|_p+\|z\|_p-\|x+y+z\|_p$ be the triple deficit, and $\mathrm{pairGapSum}(x,y,z)$ the sum of the three pair deficits $\|u\|_p+\|v\|_p-\|u+v\|_p$ over $\{u,v\}=\{x,y\},\{x,z\},\{y,z\}$. A real $C$ is an admissible Hlawka constant for $\|\cdot\|_p$ when $\mathrm{tripleGap}(x,y,z)\le C\cdot\mathrm{pairGapSum}(x,y,z)$ for all $x,y,z$.

The theorem states that for every real exponent $p>1$,

This is a coarse, purely exponent-dependent upper bound on $K_p$, valid for every real $p>1$. It complements the separate results that $K_p$ is admissible for complex diagonal triples in every finite dimension when $p\ge256$ and that every admissible constant in some dimension $n\ge3$ is at least $K_p$.