Gaussian Volume Bounds Under Hypercube Translations and Generalizations

We derive bounds for the Gaussian volume of the Minkowski sum \(Ω + K_a\), where \(Ω\) is any Borel set and \(K_a\) represents the vertices of a hypercube with edge length \(2a\). For \(a = c/√ln n\), we show that if \(c < π/√2\), then the volume remains close to that of \(Ω\). Conversely, if \(c > π\), we construct a domain \(Ω\) for which the volume of the sum is significantly less than that of \(Ω\). These results extend to generalized hypercubes, providing precise volume bounds based on the sparsity ratio of the vertices used in the translations. The findings have implications for adversarial hypothesis testing, particularly in scenarios where an adversary, after observing a Gaussian random vector, perturbs a subset of its components by a fixed magnitude.