Anderson–Klee's principle for objects in a common geometry

Let $X\subset\mathbb{E}^n$ be an object, and let $X\langle r\rangle$ denote the set of $r$-singular points, where an $r$-singular point is one whose associated fiber has scheme-theoretic linear span of dimension at least $r$. Anderson–Klee's principle. One has

$$\dim X\langle r\rangle\leq n-r-1.$$

Here dimension is understood as Hausdorff dimension in convex geometry and algebraic dimension in algebraic geometry. The assertion is known for convex geometry by work of Anderson and Klee, while its algebraic-geometric formulation is presented as conjectural.