Bounds on the radius of black hole shadows in n-dimensional Einstein gravity

The dark shadow cast by a black hole, determined by the outermost unstable null circular geodesics (the photon sphere), provides a direct probe of strong-field gravity. In this work, we derive model-independent lower and upper bounds on the shadow radius $r_{\mathrm{sh}}$ for static, spherically symmetric, asymptotically flat black holes in $n$-dimensional ($n\ge 4$) Einstein gravity, supported by an anisotropic matter field. For the lower bound, assuming the matter satisfies the Weak Energy Condition (WEC), we prove $r_{\mathrm{sh}}\geq \bigl(\frac{n-1}{2}\bigr)^{\frac{1}{n-3}}\sqrt{\frac{n-1}{n-3}}\,r_H$, where $r_H$ is the horizon radius. For the upper bound, under the WEC and the Strong Energy Condition (SEC), together with an asymptotic decay condition on the matter fields, we prove $r_{\mathrm{sh}}\leq\sqrt{\frac{n-1}{n-3}}\bigl[(n-1)M\bigr]^{\frac{1}{n-3}}$, where $M$ is the ADM mass. These results reduce to the known four-dimensional bounds and are saturated by the vacuum Schwarzschild-Tangherlini black hole. Our results generalize the four-dimensional shadow bounds to an arbitrary number of dimensions and provide model-independent geometric constraints on the observable shadow of higher-dimensional black hole spacetimes.