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i tested your code (with a small modification) with the image below (OpenCV version is 3.2).

as you can see on the result image it works as expected. probably you are using an old version of OpenCV and getting a buggy result. (i think it was a bug recently fixed)

#include "opencv2\opencv.hpp"

using namespace cv;
using namespace std;

int main() {
    //VideoCapture cap(0);
    Mat src, gray, background, binary, diff;
    //cap >> background;
    //cvtColor(background, background, CV_BGR2GRAY);
    vector<vector<Point> > contours;
    vector < vector<int> > hullI = vector<vector<int> >(1);
    vector < vector<Point> > hullP = vector<vector<Point> >(1);
    vector<Vec4i> defects;
        src = imread("hand.png");
        cvtColor(src, gray, CV_BGR2GRAY);
        blur(gray, gray, Size(3, 3));
        threshold(gray, binary, 150, 255, THRESH_BINARY_INV);
        //erode(binary, binary, Mat(Size(5, 5), CV_8U));
        imshow("binary", binary);
        findContours(binary, contours, CV_RETR_EXTERNAL, CV_CHAIN_APPROX_NONE);
        if (!contours.empty()) {
            sort(contours.begin(), contours.end(), [](vector<Point> a, vector<Point> b) { return a.size() > b.size(); });
            drawContours(src, contours, 0, Scalar(255, 255, 0));

            convexHull(contours[0], hullI[0]);
            convexHull(contours[0], hullP[0]);
            drawContours(src, hullP, 0, Scalar(0, 255, 255));

            if (hullI[0].size() > 2) {
                convexityDefects(contours[0], hullI[0], defects);

                for (Vec4i defect : defects) {
                    line(src, contours[0][defect[0]], contours[0][defect[2]], Scalar(0, 0, 255));
                    line(src, contours[0][defect[1]], contours[0][defect[2]], Scalar(0, 0, 255));
                }
            }
        }
        imshow("result", src);
        char key = waitKey(0);
        return 0;
    }