Horizonless ultra-compact objects with light rings and monotonically decreasing density or radial pressure satisfy C >= 1/3.
Extremal black holes have external light rings
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It is proved that spherically symmetric extermal black holes possess at least one external light ring. Our remarkably compact proof is based on the dominant energy condition which characterizes the external matter fields in the non-vacuum extremal black-hole spacetimes.
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A Compact theorem on the compactness of ultra-compact objects with monotonically decreasing matter fields
Horizonless ultra-compact objects with light rings and monotonically decreasing density or radial pressure satisfy C >= 1/3.