This question is related to the following: Prove/disprove convexity of surface volume in terms of a deformation field - the context is summarized below for clarity.
In the setting of convex optimization, I am looking for a convex relaxation of a potential $U=pV$ of a scalar pressure $p$ acting over some volume $V$ enclosed by a parametric surface embedded in a Cartesian frame. Hence I need a convex relaxation of $V$.
Consider the following functions: $$ \begin{split} \mathbf{X}=\mathbf{X}(\xi^1, \xi^2)\\ \\ \mathbf{u}=\mathbf{u}(\xi^1, \xi^2)\\ \\ (\xi^1, \xi^2) \in \mathbb{R}^2, \quad \mathbf{X,u} \in \mathbb{R}^3\\ \end{split} $$
Here $\mathbf{X}$ are the coordinates of a point on the undeformed surface $S$ and $\mathbf{u}$ is the deformation field, such that, using convected coordinates, the position of the original point on the deformed surface $s$ is given by $\mathbf{x=X+u}$.
The enclosed volume $V$ of $s$ can now be defined by application of the divergence theorem by integrating a function of unit divergence: $$ \begin{split} V=\frac{1}{3}\int{\mathbf{x}\cdot \mathbf{n}} da \end{split}. $$
As shown in my related question, this integral can be written as: $$ \begin{split} V = \frac{1}{3}\int{ \mathbf{x} \cdot \left((\frac{\partial \mathbf{X}}{\partial \xi^1} + \frac{\partial \mathbf{u}}{\partial \xi^1}) \times (\frac{\partial \mathbf{X}}{\partial \xi^2} + \frac{\partial \mathbf{u}}{\partial \xi^2}) \right)} d\xi^1d\xi^2 \end{split}. $$ In the above, the vector cross product represents the (outward oriented) non-unit normal to the deformed surface $s$, which is the cross product of the tangent vectors on $s$, say $\mathbf{g_1}$ and $\mathbf{g_2}$. Thus $V$ can be interpreted as the sum of the volumes of differential pyramids of height $\mathbf{x}$ and (oriented) differential base area $(\mathbf{g_1 \times g_2}) d\xi^1d\xi^2$.
Now assuming the origin is always 'inside' the surface, each differential pyramid should always have positive volume, which leads me to believe the following is a suitable convex relaxation of $V$:
$$ \begin{split} U = pV = p\int _V dV = \frac{p}{3}\int \left| (\mathbf{X+u}) \cdot \left[\bigg(\frac{\partial \mathbf{X}}{\partial \xi^1} + \frac{\partial \mathbf{u}}{\partial \xi^1}\bigg) \times \bigg(\frac{\partial \mathbf{X}}{\partial \xi^2} + \frac{\partial \mathbf{u}}{\partial \xi^2}\bigg) \right] \right| d\xi^1d\xi^2 \end{split} $$
Is the above integral always convex in $\mathbf{u}$ for a fixed $\mathbf{X}$, or are further restrictions needed? Does the surface itself need to be convex?