L^p-Mahler conjecture for symmetric convex bodies

Establish that for every origin-symmetric convex body K ⊂ R^n and every p > 0, the L^p-Mahler volume M_p(K) = ∫_{R^n} |K| e^{-h_{p,K}(y)} dy, where h_{p,K}(y) = (1/p) log ∫_K e^{p⟨x,y⟩} dx, satisfies M_p(K) ≥ M_p(B_2^n).

Background

The authors introduce an L^p analogue of the Mahler volume using the p-support function h_{p,K}(y) = (1/p) log ∫_K e^{p⟨x,y⟩} dx. The resulting quantity M_p(K) is GL(n,R)-invariant and interpolates structures linked to both convex and complex techniques.

Conjecture 8.3 posits that, among symmetric convex bodies, the Euclidean ball B_2^n minimizes M_p, paralleling the classical Mahler conjecture in the p=1 case and providing a bridge for complex-analytic approaches to Mahler-type inequalities.