Higher traces of linear maps on finite-dimensional normed spaces

We prove a unified trace-average formula for the $k$-th higher trace ${\lambda }_k(A)=\mathrm{tr}({\mathrm{\Lambda }}^{k}A)$ of a linear operator $A$ on a finite-dimensional normed space. The formula averages the matrix coefficient $w\mapsto ⟨({\mathrm{\Lambda }}^{k}A)w,w^{\ast }⟩$ over the unit sphere of ${\mathrm{\Lambda }}^{k}X$ against a probability measure $\eta$; it holds for \emph{all} $A$ if and only if the operator-valued average $T_{\eta }=\left(\genfrac{}{}{0px}{}{N}{k}\right)\int w\otimes w^{\ast }\mathrm{d}\eta$ 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 ${\mathrm{\Lambda }}^k{\mathbb{R}}^N$; 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.