SL n Contravariant Matrix-Valued Valuations on Polytopes

Abstract

Without any continuity assumptions, a complete classification of $\textrm{SL}(n)$ contravariant, matrix-valued valuations on convex polytopes is established. Furthermore, the constraint for matrix symmetry is removed. If $n\geq 4$, then such valuations are uniquely characterized by the generic Lutwak–Yang–Zhang matrix; in dimension three, a new function appears. The classification result in the 2-dimensional case is consistent with the established example of $\textrm{SL}(2)$-equivariant matrix-valued valuation.