Equality characterization in Lemma 23: simplices as unique extremizers

Characterize the equality cases in the Lp Rogers–Shephard-type inequality for locally anti-blocking bodies established in Lemma 23 by proving that equality |K ⊕_p K| = κ_{n,q} |K| holds if and only if K is a simplex.

Background

Lemma 23 provides a sharp constant κ{n,q} for the inequality |K ⊕_p K| ≤ κ{n,q} |K| for locally anti-blocking bodies, with κ_{n,q} expressed via generalized binomial coefficients. While the constant is shown to be sharp (e.g., achieved by simplices), the complete characterization of all equality cases is not established.

The authors explicitly formulate a conjecture asserting that simplices are the only bodies achieving equality, inviting a definitive classification of extremizers under the locally anti-blocking condition.