Symmetric Mahler conjecture

Take a rectangle centered at the origin, four units wide and two units tall. Its polar is a diamond with corners half a unit to the left and right, and one unit up and down. The rectangle has area 8 and the diamond area 1, so their product is 8. The polar collects coordinate weights whose weighted sums never exceed 1 on the original shape; a wider rectangle makes the diamond narrower. The conjecture asks how small this volume product can be for any solid that is convex and symmetric through the origin, in four or more dimensions.[1]

A solid shape constrains which weighted sums of coordinates stay at most 1 throughout it. How small can the product of the shape's volume and the volume of this set of allowed weights be, in dimension four or higher? The question concerns a body symmetric through the origin and its polar. Here a convex body $K$ is closed and bounded with nonempty interior, and contains every segment between its points. Symmetry means $K=-K$, with the origin inside. A measurement vector $y$ belongs to the polar $K^\circ$ when $x\cdot y\leq1$ for every point $x$ of $K$. The dot product $x\cdot y$ multiplies corresponding coordinates and adds them.

For the planar rectangle $[-2,2]\times[-1,1]$, the largest pairing with $(u,v)$ is $2|u|+|v|$. Its polar therefore obeys $2|u|+|v|\leq1$: the diamond with diagonals of lengths 1 and 2. The areas are 8 and 1, giving product 8. This is a checked two-dimensional illustration, not evidence for all higher-dimensional bodies.

Let $P(K)$ denote the body's volume times its polar's volume. In dimension $n\geq4$, the conjecture is $$P(K)\geq\frac{4^n}{n!}.$$ Here $n!$ is the product of the integers from 1 through $n$. An invertible linear change of coordinates multiplies the body's volume by one constant factor and the polar's by the reciprocal, leaving the product unchanged. This lets the question concern shape rather than a chosen linear coordinate system.[1]

The proposed answer is proved up to three dimensions, but not for every higher dimension.[2]

The established three-dimensional theorem does not extend automatically to n≥4. The nonsymmetric problem has a different bound.[2]

Stretching the rectangle increases its area but shrinks its polar by the reciprocal factor, so the product stays the same. This makes the product a way to compare shapes and their measurement constraints without rewarding a mere change of scale. The conjecture seeks the smallest possible value of that comparison.[1]

Polarity turns a set of points into the collection of linear inequalities that bound it. The conjecture would give a universal sharp tradeoff between the size of a symmetric body and the size of these dual constraints. Because the product is unchanged by invertible linear coordinate changes, the tradeoff captures geometric structure that individual volumes miss. The established three-dimensional result supplies such a sharp comparison in one dimension range; the general higher-dimensional target asks for the same structural limit.[1]