# Right-triangle distance inequality in a product space

Let $$(C_i)_{i=1}^m$$ be nonempty closed convex subsets of a real Hilbert space $$\mathcal{H}$$. I am interested in proving (or finding a counterexample to) the following conjecture on the $$m$$-fold product space $$H=\mathcal{H}\times\cdots\times\mathcal{H}$$ equipped with the inner product $$\langle x\,|\,y\rangle=\sum_{i=1}^m\omega_i\langle x_i\,|\,y_i\rangle$$ (for convex weights $$\{\omega_i\}_{i=1}^m\subset]0,1]$$ satisfying $$\sum_{i=1}^m\omega_i=1$$). For convenience, set $$D=\{x\in H\,|\,(\forall i,j\in\{1,\ldots,m\})\quad x_i=x_j\}$$ (sometimes called the diagonal subspace).

The conjecture is the following inequality of distance functions

$$(\forall x\in \times_{i=1}^m C_i)\quad \textrm{dist}_{\bigcap_{i=1}^mC_i}\left(\sum_{i=1}^m\omega_i x_i\right)\leq\textrm{dist}_{D}(x).$$

Based on the geometry, it looks like as soon as this inequality is violated, it necessitates that at least one component of $$x$$ is no longer in $$C_i$$. However, I have not been able to prove this by contradition yet.

These two distances form the the smaller legs of a right triangle, with endpoints $$x$$, $$P_Dx$$, and $$P_{D\cap\times_{i=1}^m(\bigcap_{j=1}^mC_j)}x$$. It is a right triangle because the leg $$x-P_Dx$$ (whose magnitude is $$\textrm{dist}_D(x)$$) is orthogonal to $$P_Dx-P_{D\cap\times_{i=1}^m(\bigcap_{j=1}^mC_j)}x$$ (whose magnitude is $$\textrm{dist}_{\bigcap_{i=1}^mC_i}\left(\sum_{i=1}^m\omega_i x_i\right)$$).

Here is an (updated) example graph -- the conjecture is saying that, as long as $$x\in \times_{i=1}^m C_i$$, the length of the segment between $$x$$ and the projection $$P_Dx$$ is always larger than the length of the segment connecting $$P_Dx$$ to $$P_{(C_1\cap C_2)\times(C_1\cap C_2)}(P_Dx)$$.

It may be convenient to note that the projection onto $$D$$ is given by the average repeated in every component, i.e., $$P_Dx=(\sum_{i=1}^m\omega_i x_i)_{j\in\{1,\ldots,m\}}$$, so $$\textrm{dist}^2_D(x)=\sum_{i=1}^m\omega_i\|x_i-\sum_{j=1}^m\omega_jx_j\|^2$$. Even if we could prove (or disprove) this for $$\omega_i\equiv 1/m$$, I would be happy.

The conjecture is false, even in $$2$$D:

Consider two skew lines, $$\omega_1=\omega_2=1/2$$.

$$C_1=\{(x,y)\in\mathbb{R}^2\,|\,y=0.1x-1\}\quad C_2=\{(x,y)\in\mathbb{R}^2\,|\,y=-0.1+1\}.$$

Then $$x=((0,1),(0,-1))\in C_1\times C_2$$. However, $$\textrm{dist}_{D}(x)=\|P_Dx-x\|=2$$ while $$\textrm{dist}_{C_1\cap C_2}(\frac{x_1+x_2}{2})=10$$.