  For a broad class of planar Markov processes, viz. L\'evy processes
satisfying certain conditions (valid \textit{eg} in the case of Brownian motion
and L\'evy flights), we establish an exact, universal formula describing the
shape of the convex hull of sample paths. We show indeed that the average
number of edges joining paths' points separated by a time-lapse $\Delta \tau
\in \left[\Delta \tau _1, \Delta \tau_2\right]$ is equal to $2\ln \left(\Delta
\tau_2 / \Delta \tau_1 \right)$, regardless of the specific distribution of the
process's increments and regardless of its total duration $T$. The formula also
exhibits invariance when the time scale is multiplied by any constant.
  Apart from its theoretical importance, our result provides new insights
regarding the shape of two-dimensional objects modelled by stochastic
processes' sample paths (\textit{eg} polymer chains): in particular for a total
time (or parameter) duration $T$, the average number of edges on the convex
hull ("cut off" to discard edges joining points separated by a time-lapse
shorter than some $\Delta \tau < T$) will be given by $2 \ln
\left(\frac{T}{\Delta \tau}\right)$. Thus it will only grow logarithmically,
rather than at some higher pace.
