What is the most likely failure point?

Definition

The most likely failure point is also known as design point in structural reliability analysis. It is the point in the failure domain that has the smallest distance to the origin in the underlying standard Normal space; i.e.:
$$
\mathbf{u}^* = \underset{\mathbf{u}\in\mathbf{U}_\mathcal{F}}{\operatorname{arg\,min}}\left(\|\mathbf{u}\|\right)\;,
$$ where $\mathbf{u}^*$ is the design point in standard Normal space and $\mathbf{U}_\mathcal{F}=\left\{\mathbf{u}\in\mathrm{R}^N|G(\mathbf{u})\le0\right\}$ the failure domain in standard Normal space.

The coordinates of the design point $\mathbf{x}^*$ in the original space are then obtained by
$$
\mathbf{x}^* = T^{-1}(\mathbf{u}^*)\;,
$$ where $T^{-1}(\cdot)$ is the transformation from the standard Normal space to the original space.

Interpretation

Of all the points in the failure domain, the design point corresponds to the most likely explanation of failure — therefore the name "most likely failure point". Generally, the area around the design point contributes most to the integral of the probability of failure. The design point is thus a logical choice for the reference point of the system of interest. The popular first-order reliability method (FORM) linearizes the limit state function around this point.