Summary: An implicit assumption in the original version of Holditch's theorem is \(C^1\)-regularity and strict convexity of the envelope generated by a chord traveling around a convex curve \(\mathbb{C}\). We establish that this holds when \(\mathbb{C}\) is \(C^2\)-regular with positive curvature and the chordlength is sufficiently small. We also consider the case where \(\mathbb{C}\) is polyhedral. Then, strict convexity of the envelope may not hold, but, for sufficiently small chordlength, it is nevertheless \(C^1\)-regular. The case of a general convex curve \(\mathbb{C}\) remains an open problem.