Return to Question

Setting:

Suppose $\{u_i\}_{i=1}^n \subset R^d$ is a collection of unit vectors such that $u_i^Tu_j < 0$ for all $i\neq j$, and $w$ is a unit vector such that $u_i^T w> 0$ for all $i=1,\dotsc,n$. Let $a \in R^d$ be a unit vector.

Goal:

We want to show that the following hold or to find a counter-example.

$$\sum_{i=1}^n \lvert u_i^T a\rvert u_i^T w < 1$$

We can prove this statement for the case of $n=2$, but we were not able to prove it for $n>2$.

Any suggestions?

Setting:
Suppose $\{u_i\}_{i=1}^n \subset R^d$ is a collection of unit vectors such that $u_i^Tu_j < 0$ for all $i\neq j$, and $w$ is a unit vector such that $u_i^T w> 0$ for all $i=1,\dotsc,n$. Let $a \in R^d$ be a unit vector.

Goal:
We want to show that the following hold or to find a counter-example.

$$\sum_{i=1}^n \lvert u_i^T a\rvert u_i^T w < 1$$

We can prove this statement for the case of $n=2$, but we were not able to prove it for $n>2$.

Any suggestions?

Update:
We used Echo's approach from the comment. It worked for the first iteration (when the local maximum is an interior point) from the stationarity condition of the gradient. However, we got stuck at the second iteration (when one of the constraints was active). So now we have a stationarity condition on the projected gradient. We couldn't solve this. Any help would be appreciated.

Setting:

Suppose $\{u_i\}_{i=1}^n \subset R^d$ is a collection of unit vectors such that $u_i^Tu_j < 0$ for all $i\neq j$, and $w$ is a unit vector such that $u_i^T w> 0$ for all $i=1,\dotsc,n$. Let $a \in R^d$ be a unit vector.

Goal:

We want to show that the following hold or to find a counter-example.

$$\sum_{i=1}^n \lvert u_i^T a\rvert u_i^T w < 1.$$

We can prove this statement for the case of $n=2$, but we were not able to prove it for $n>2$.

Any suggestions?

Setting:

Suppose $\{u_i\}_{i=1}^n \subset R^d$ is a collection of unit vectors such that $u_i^Tu_j < 0$ for all $i\neq j$, and $w$ is a unit vector such that $u_i^T w> 0$ for all $i=1,\dotsc,n$. Let $a \in R^d$ be a unit vector.

Goal:

We want to show that the following hold or to find a counter-example.

$$\sum_{i=1}^n \lvert u_i^T a\rvert u_i^T w < 1$$

We can prove this statement for the case of $n=2$, but we were not able to prove it for $n>2$.

Any suggestions?

Setting:

Suppose $\{u_i\}_{i=1}^n \subset R^d$ is a collection of unit vectors such that $u_i^Tu_j < 0$ for all $i\neq j$, and $w$ is a unit vector such that $u_i^T w> 0$ for all $i=1,...,n$. Let $a \in R^d$ be a unit vector.

Goal:

We want to show that the following hold or to find a counter-example.

$\sum_{i=1}^n |u_i^T a| u_i^T w < 1$.

We can prove this statement for the case of n=2, but we were not able to prove it for n>2.

Any suggestions?

Setting:

Suppose $\{u_i\}_{i=1}^n \subset R^d$ is a collection of unit vectors such that $u_i^Tu_j < 0$ for all $i\neq j$, and $w$ is a unit vector such that $u_i^T w> 0$ for all $i=1,\dotsc,n$. Let $a \in R^d$ be a unit vector.

Goal:

We want to show that the following hold or to find a counter-example.

$$\sum_{i=1}^n \lvert u_i^T a\rvert u_i^T w < 1.$$

We can prove this statement for the case of $n=2$, but we were not able to prove it for $n>2$.

Any suggestions?