On the Areas of Level Sets in Compact Connected Sublattices of Three-Dimensional Euclidean Space

G. Gierz

Journal of Lie theory, Tome 30 (2020) no. 2, pp. 371-405

Résumé

As is well-known the three-dimensional Euclidean space $\Re ^{3}$, equipped with the order relation $\left (x_{1} ,x_{2} ,x_{3}\right ) \leq \left (x_{1}^{ \prime } ,x_{2}^{ \prime } ,x_{3}^{ \prime }\right )$ if $x_{i} \leq x_{i}^{ \prime }$ for $i =1 ,2 ,3\text{,}$ is a distributive, topological lattice. Let $L$ be a compact, connected sublattice of $\Re ^{3}\text{.}$ For $\left (x_{1} ,x_{2} ,x_{3}\right ) \in L$ we define $\lambda \left (x_{1} ,x_{2} ,x_{3}\right ) =x_{1} +x_{2} +x_{3}$ and for $r \in \Re $ we let $L_{r} =\left \{\left (x_{1} ,x_{2} ,x_{3}\right ) \in L :\lambda \left (x_{1} ,x_{2} ,x_{3}\right ) =r\right \}$. If $\mu_{L} \left (r\right )$ denotes the surface area of $L_{r}\text{,}$ then we show that the function $r \mapsto \mu _{L} \left (r\right )$ is continuously differentiable, and that the value of $\mu _{L}^{ \prime } \left (r\right )$ can be computed in two different ways: Either as an integral of a certain function over the boundary of $L_{r}\text{,}$ or as the value of the expression $\sqrt{3} \left (\lambda \left (\sup L_{r}\right ) +\lambda \left (\inf L_{r}\right ) -2 r\right )$.

Détail

Classification : 06B30, 26B20, 54F05, 26B15
Mots-clés : Level sets and rank functions, sublattices of R³, integral formulas