  This paper addresses a two-player target defense game in the $n$-dimensional
Euclidean space where an attacker attempts to enter a closed convex target set
while a defender strives to capture the attacker beforehand. We provide a
complete and universal differential game-based solution which not only
encompasses recent work associated with similar problems whose target sets have
simple, low-dimensional geometric shapes, but can also address problems that
involve nontrivial geometric shapes of high-dimensional target sets. The value
functions of the game are derived in a semi-analytical form that includes a
convex optimization problem. When the latter problem has a closed-form
solution, one of the value functions is used to analytically construct the
barrier surface that divides the state space of the game into the winning sets
of players. For the case where the barrier surface has no analytical expression
but the target set has a smooth boundary, the bijective map between the target
boundary and the projection of the barrier surface is obtained. By using
Hamilton-Jacobi-Isaacs equation, we verify that the proposed optimal state
feedback strategies always constitute the game's unique saddle point whether or
not the optimization problem has a closed-form solution. We illustrate our
solutions via numerical simulations.
