>If true, you're very correct that lower-dimensional intuition does not transfer into higher-dimensional spaces: my intuition tells me that a Gaussian distribut

>If true, you're very correct that lower-dimensional intuition does not transfer into higher-dimensional spaces: my intuition tells me that a Gaussian distribution drops off as you fall away from the mean, and it's quite easy for me to imagine that in 2 dimensions, 3 dimensions (e.g. by imagining a mound on a plane) and 4 dimensions (e.g. a cloud in 3-space with increased density around the mean). Density is different from mass. Namely, mass is the integral of density. So your intuition is roughly correct for density, but you need to make it accord with a good intuition for mass. Since getting the mass requires an integral, getting the mass over N-dimensional distributions requires integrating an N-dimensional region, which means N integrations for N dimensions. Each integration is, intuitively, a kind of sum. Integrating out many dimensions happens recursively; looped or recursive addition is multiplication. So on some level, to take the probability mass of a region in N-dimensional space, you need to "multiply" a density. Since the total probability mass is fixed (1.0), adding more dimensions means you need to "multiply" the density by a larger number to get the mass, which means you need to divide the mass by a larger number to get the density, which means that despite the density peaking at the mean, the available density at any given point gets smaller as the dimensionality rises.