Subsets of a Real Hilbert Space Translation-Closed along an Orthonormal Tuple

In the settings of Real Hilbert Spaces: Standing Notation and Background and Euclidean Space and Real Matrices: Standing Notation, the latter used in the dimension $m$ for a natural number $m$ with $1\le m$. Let $H$ be a real Hilbert space, with its inner product and its sums of vectors as fixed there, and let $\mathbb{R}^{m}$ be as fixed in Euclidean Space and Real Matrices: Standing Notation (i). Let $e\in H^{m}$ be an $m$-tuple in $H$ that is orthonormal, and let $\Lambda^{\sharp}:\mathbb{R}^{m}\to H$ be the map determined by $e$ in Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions (i).

(Translation-closed along a tuple)(i) A subset $A\subseteq H$ is translation-closed along $e$ if it is nonempty and

$$x+\Lambda^{\sharp}\zeta\in A\qquad\text{for every }x\in A\text{ and every }\zeta\in\mathbb{R}^{m}.$$