IMC 2024 · Day 1 · P5

By radial symmetry of the distribution, $f(p)$ depends only on $|op|$ (the distance between $o$ and $p$), so, we may assume that $p$ lies on the segment between $o$ and $q$. For points $x_1, \dots, x_n$ and $x \in B$ denote by $f_x(x_1, \dots, x_n)$ the indicator function of the event “$x$ is in the convex hull of $x_1, \dots, x_n$”. The claim follows from the following deterministic inequality $$\tag{1} \sum f_p(\pm x_1, \dots, \pm x_n) \geqslant \sum f_q(\pm x_1, \dots, \pm x_n),$$ where $x_1, \dots, x_n \in B$ are arbitrary points in general position and the summations are over all $2^n$ choices of signs (here $o$ is identified with the origin, that is, $x$ and $-x$ are symmetric with respect to $o$). Indeed, taking the expectation in (1) over independent random uniform $x_1, \dots, x_n$, we get $2^n f(p) \geqslant 2^n f(q)$. (To be specific, here “general position” means that for any point set $A \subset \{\pm x_1, \dots, \pm x_n, p, q\}$, which does not contain simultaneosuly $x_i$ and $-x_i$, is not contained in an (affine) $(|A| - 2)$-dimensional plane. This holds with probability 1.)

To prove (1), we use the following formula for the characteristic function $\chi_P$ of the convex polyhedron $P \subset \mathbb{R}^d$: if $P_1, \dots, P_k$ are all facets of $P$, and $Q_i$ is the convex hull of $o$ and $P_i$, then $\chi_P = \sum \pm \chi_{Q_i}$, where the sign is plus if $o$ and $P$ are on the same side of $P_i$, and minus otherwise. Indeed, for every point $p$ in general position look how the ray $op$ intersects the boundary of $P$ and realize that for at most two summands the contribution of the RHS at point $p$ is non-zero, and the total contribution equals 1 when $p$ is inside $P$ and 0 (possibly as $0 = 1 - 1$) otherwise. Use this formula for every polyhedron $P$ with $n$ vertices $y_1, \dots, y_n$, where each $y_i$ is $\pm x_i$. These polyhedrons are simplicial (all facets are simplices) because of the general position condition. Sum up over all $2^n$ such $P$, we get the expression of $\sum_P \chi_P$ as a linear combination of $\chi_S$, where $S$ are simplices formed by $o$ and some $d$ points in $\{\pm x_1, \dots, \pm x_n\}$ (not containing $x_i$ and $-x_i$ simultaneously).

For proving (1), it suffices to verify that all coefficients of $\chi_S$ in this linear combination are positive (since two sides of (1) are the values of the sum $\sum_P \chi_P$ at $p$ and $q$). Let's find a coefficient of $\chi_S$, where, say, $S$ is a simplex with vertices $o, x_1, \dots, x_d$. The plane $\alpha$ through $x_1, \dots, x_d$ partitions $\mathbb{R}^d$ onto two parts $H^+$ (containing $o$) and $H^-$ (not containing $o$). For every pair $\{x_i, -x_i\}$ with $i > d$, either both points belong to $H^+$, or one belongs to $H^-$ and another to $H^+$. $\chi_S$ goes with the plus sign for $P$ with vertices $x_1, \dots, x_d$ and other vertices from $H^+$, and with the minus sign for $P$ with vertices $x_1, \dots, x_d$ and other vertices from $H^-$. It is immediate that there are at least as many pluses as minuses.