> For the complete documentation index, see [llms.txt](https://jimmylin1991.gitbook.io/practice-of-algorithm-problems/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://jimmylin1991.gitbook.io/practice-of-algorithm-problems/math/469.-convex-polygon.md).

# 469. Convex Polygon

Given a list of points that form a polygon when joined sequentially, find if this polygon is convex [(Convex polygon definition)](https://en.wikipedia.org/wiki/Convex_polygon).

**Note:**

1. There are at least 3 and at most 10,000 points.
2. Coordinates are in the range -10,000 to 10,000.
3. You may assume the polygon formed by given points is always a simple polygon[ (Simple polygon definition)](https://en.wikipedia.org/wiki/Simple_polygon). In other words, we ensure that exactly two edges intersect at each vertex, and that edges otherwise **don't intersect each other**.

**Example 1:**

```
[[0,0],[0,1],[1,1],[1,0]]

Answer: True

Explanation:
```

**Example 2:**

```
[[0,0],[0,10],[10,10],[10,0],[5,5]]

Answer: False

Explanation:
```

```cpp
int crossProduct(const vector<int>& A, const vector<int>& B, const vector<int>& C) {
    int BAx = B[0] - A[0];
    int BAy = B[1] - A[1];
    int CBx = C[0] - B[0];
    int CBy = C[1] - B[1];
    return BAx * CBy - BAy * CBx;
}
bool isConvex(vector<vector<int>>& points) { // time: O(n); space: O(1)
    int n = points.size(), j, k, cross;
    bool positive_cross = false, negative_cross = false;
    for (int i = 0; i < n; ++i) {
        j = (i + 1) % n;
        k = (j + 1) % n;
        cross = crossProduct(points[i], points[j], points[k]);
        if (cross > 0) positive_cross = true;
        else if (cross < 0) negative_cross = true;
        if (positive_cross && negative_cross) return false;
    }
    return true;
}
```