We establish a connection between linear complementary dual (LCD) codes and caps in projective space. Using this framework and the structure theory of maximal binary caps, we derive nonexistence theorems for binary LCD codes with minimum distance at least $4$, providing computation-free proofs that were previously obtained only through exhaustive search. As an application, we completely determine the optimal minimum distances for codimensions $7$ and $8$ for the first time. The framework applies to ternary LCD codes and quaternary Hermitian LCD codes, where it reduces the problem to the structure of large caps. Using known results on large caps, we determine the optimal minimum distances of ternary LCD codes of codimension $6$ for all lengths, and we explain why the phenomena behind the binary nonexistence theorems, such as the dependence on the parity of the length, have no counterpart for these codes.

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