Riesz rising sun lemma for several variables and the John–Nirenberg inequality

We obtain a multidimensional analog of the well-known Riesz rising sun lemma. We prove a more precise version of this lemma for space dimension $d=2$. We use these lemmas to establish an anisotropic analog of the John–Nirenberg inequality for functions of bounded mean oscillation with an exact constant in the exponent. Earlier, this exact constant was only known in the one-dimensional case.