As a guideline for experimental tests of the ideal glass transition (random-pinning glass transition, RPGT) that shall be induced in a system by randomly pinning particles, we performed first-principle computations within the hypernetted chain approximation and numerical simulations of a hard-sphere model of a glass former. We obtain confirmation of the expected enhancement of glassy behavior under the procedure of random pinning. We present the analytical phase diagram as a function of c and of the packing fraction ϕ, showing a line of RPGT ending in a critical point. We also obtain microscopic results on cooperative length scales characterizing medium-range amorphous order in hard-sphere glasses and indirect quantitative information on a key thermodynamic quantity defined in proximity to ideal glass transitions, the amorphous surface tension. Finally, we present numerical results of pair correlation functions able to differentiate the liquid and the glass phases, as predicted by the analytic computations.