  This paper introduces a new multidimensional extension of the
Hegselmann-Krause (HK) opinion dynamics model, where opinion proximity is not
determined by a norm or metric. Instead, each agent trusts opinions within the
Minkowski sum $\xi+\mathcal{O}$, where $\xi$ is the agent's current opinion and
$\mathcal{O}$ is the confidence set defining acceptable deviations. During each
iteration, agents update their opinions by simultaneously averaging the trusted
opinions. Unlike traditional HK systems, where $\mathcal{O}$ is a ball in some
norm, our model allows the confidence set to be non-convex and even unbounded.
  We demonstrate that the new model, referred to as SCOD (Set-based Confidence
Opinion Dynamics), can exhibit properties absent in the conventional HK model.
Some solutions may converge to non-equilibrium points in the state space, while
others oscillate periodically. These ``pathologies'' disappear if the set
$\mathcal{O}$ is symmetric and contains zero in its interior: similar to the
usual HK model, SCOD then converges in a finite number of iterations to one of
the equilibrium points. The latter property is also preserved if one agent is
"stubborn" and resists changing their opinion, yet still influences the others;
however, two stubborn agents can lead to oscillations.
