Makeev conjectured that every constant-width body is inscribed in thedual difference body of a regular simplex. We prove that homologically,there are an odd number of such circumscribing bodies in dimension3, and therefore geometrically there is at least one. We show thatthe homological answer is zero in higher dimensions, a result whichis inconclusive for the geometric question. We also give a partialgeneralization involving affine circumscription of strictly convex bodies.
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