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How large are the smallest-area projections of a high-dimensional convex body?

Let $B$ be a convex body in $\mathbb{R}^d$, equipped with its standard Euclidean form, and assume that $$\intop_B x \, dx = 0$$ $$\frac{1}{|B|_d} \intop_B x_i x_j \, dx = \delta_{ij},$$ a normalizat …