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Let n be a positive integer, λ0 a real number, and 1≤ p≤ ∞. We study the unit disk random geometric graphGp(λ,n), defined to be the random graph on n vertices, independently distributed uniformly in the standard unit disk in ${\mathbb R}^2$, with two vertices adjacent if and only if their ℓp-distance is at most λ. Let $\lambda=c\sqrt{\ln n/n}$, and let ap be the ratio of the (Lebesgue) areas of the ℓp- and ℓ2-unit disks. Almost always, Gp(λ,n) has no isolated vertices and is also connected if cap−−1/2, and has $n^{1-a_pc^2}(1+o(1))$ isolated vertices if cap−−1/2. Furthermore, we find upper bounds (involving λ but independent of p) for the diameter of Gp(λ,n), building on a method originally due to M. Penrose.