Linearity of the volume. Looking for a characterization of sausages

Let K,E,LK,E,L be convex bodies, dim⁡L≤1dim⁡L≤1 and K=L+EK=L+E, a sausage. In this case vol(λK+(1−λ)E)=λvol(K)+(1−λ)vol(E)vol(λK+(1−λ)E)=λvol(K)+(1−λ)vol(E). We prove that under the sole assumption that K and E   have an equal volume projection (or a common maximal volume section), if the above equality holds for just one value in (0,1)(0,1), then K=L+EK=L+E with dim⁡L≤1dim⁡L≤1. However, even having equality for all λ∈[0,1]λ∈[0,1], if no extra assumption on K,EK,E is done, such a characterization is not possible. This problem is connected with a conjecture relating the roots of the Steiner polynomial of a pair of convex bodies to their relative inradius. Counterexamples for the general case are explicitly given. In the same line, a counterexample to a conjecture by Matheron on inner parallel bodies is also shown.