A new counterexample to sangwine–Yager’s conjecture

Abstract

Sangwine-Yager conjectured in [9] that if r1 ... r n are the real parts of the roots of the (formal) alternating Steiner polynomial of V(K−tE), then 0 < r1 r(K; E) R(K; E) r n , where r(K; E) and R(K; E) are the inradius, respectively, outradius, or circumradius, of K relative to E . We present here a new counterexample to this conjecture in dimension 3 when none of the bodies is a Euclidean ball. Previous examples due to Henk and Hernández Cifre, and, respectively, to Hernández Cifre and Saorín, were constructed with fairly technical tools. Our example is non-trivial in the sense that both K and E are top dimensional convex bodies, yet it is easy to present.