Higher traces of linear maps on finite-dimensional normed spaces

We prove a unified trace-average formula for the $k$-th higher trace $\lambda_k(A)=\operatorname{tr}(\Lambda^k A)$ of a linear operator $A$ on a finite-dimensional normed space. The formula averages the matrix coefficient $w\mapsto\langle(\Lambda^kA)w, w^*\rangle$ over the unit sphere of $\Lambda^kX$ against a probability measure $\eta$; it holds for \emph{all} $A$ if and only if the operator-valued average $T_\eta=\binom{N}{k}\int w\otimes w^*\operatorname{d}\eta$ equals the identity. Two natural choices of $\eta$ satisfy this isotropy: (i) the hypersurface measure when a finite isometry group acts as an orthogonal $2$-design on $\Lambda^k\mathbb{R}^N$; and (ii) the cone probability measure (no symmetry needed). We also identify a first-order obstruction for hypersurface averages at $k=1$: only degree-$2$ spherical harmonics of the support function contribute.