Blaschke–Santaló and Mahler inequalities for the first eigenvalue of the Dirichlet Laplacian

For K belonging to the class of convex bodies in R n , we consider the λ 1 ‐product functional, defined byλ 1 ( K ) λ 1 ( K o ) , where K o is the polar body of K , andλ 1 ( · )is the first Dirichlet eigenvalue of the Dirichlet Laplacian. As a counterpart of the classical Blaschke–Santaló inequality for the volume product, we prove that the λ 1 ‐product is minimized by balls. Much more challenging is the problem of maximizing the λ 1 ‐product modulo invertible linear transformations, which is the analog of the famous Mahler conjecture for the volume product in Convex Geometry. We solve the problem in dimension n = 2 for axisymmetric convex bodies, by proving that the solution is the square. To that aim we first reduce our problem to a reverse Faber–Krahn inequality for axisymmetric convex octagons, and then we identify an optimal octagon with the one that degenerates into a square. For this latter challenge, we employ a hybrid method inspired by the Polymath blog by Tao, which is based on the joint use of theoretical arguments to settle octagons lying in computable ‘neighborhoods’ of the square, and of a numerical argument (rigorously working thanks to the monotonicity by inclusions of the involved functionals) to settle octagons lying outside the confidence zones.