Assumptions
Let
$L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric),
$a,b$ be arbitrary $n$-dimensional points,
$c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point in $L_p$-metric segment between $a$ and $b$,
$t$ be another point (possibly outside the segment).
$1 \leq p \leq \infty$
$1 \leq q \leq \infty$
$L_q(a,t)$, $L_q(b,t)$, $L_p(a,b)$ are known and finite.
The question
I'd like to estimate (tight) bounds on distance $L_q(c,t)$, preferably without referring to $a,b,c,t$ as points/vectors, instead using metrics only, however if this is necessary I'll accept that.
Are there any suggestions how to solve the above problem?
Alternative question
Alternatively to the above, it would be enough to me to show that $L_q(c,t) \leq \max\{L_q(a,t), L_q(b,t)\}$.
Note, it is easy to show that the above condition is false for $1 \leq p < q \leq \infty$, given e.g., $p=1, q=2$ (draw it in 2D). However how to prove this for $1 \leq q \leq p \leq \infty$ or designate other bounds on $p$ and $q$ for which the condition is true?
Possible hint
For $q \leq p$ the following is true (1):
$L_p(x,y) \leq L_q(x,y) \leq n^{1/q - 1/p} L_p(x,y)$
where $n$ is dimensionality of $L_p$ space.
