2 problems

Let $(U,V)$ be the coordinates of a symmetric log-concave random vector in $\mathbb{R}^2$. Ball–Nayar–Tkocz's conjecture. … This inequality is equivalent to the triangle inequality…

For $p>\frac{d}{d+2}$, define $\beta_p$ by … Let $Z^{(p)}$ be a random vector drawn from $g_{\beta_p}$. Let $X_1,\ldots,X_n$ be independent random vectors in $\mathbb{R}^d$, and le…