Step 1: The phrase perifascicular atrophy maps directly to dermatomyositis. Atrophy concentrated at the rim of each fascicle is the signature histologic clue.
Step 2: Pathophysiology - antibody and complement deposition injures the intramuscular microvasculature, and because perfusion is poorest at the fascicular border, the outermost myofibres shrink first. A perivascular mixed lymphocytic infiltrate accompanies this, and anti-Jo-1 may be positive in the antisynthetase form.
Step 3: Discard the others - steroid myopathy selectively wastes type II fibres, inclusion body myositis produces rimmed vacuoles, and nemaline myopathy produces rod-shaped inclusions.
\[\boxed{\text{Dermatomyositis}}\]