An ant lies on each corner of a $20 \times 23$ rectangle. Each second, each ant independently and randomly chooses to move one unit vertically or horizontally away from its corner. After $10$ seconds, find the expected area of the convex quadrilateral whose vertices are the positions of the ants.
1. **Understanding the Problem:**
   - We have a $20 \times 23$ rectangle with ants at each corner.
   - Each ant moves randomly either vertically or horizontally each second for 10 seconds.
   - We need to find the expected area of the convex quadrilateral formed by the ants after 10 seconds.

2. **Analyzing the Movement:**
   - Each ant moves 10 times, and each move is either vertical or horizontal.
   - Since the moves are independent and random, on average, each ant will make 5 vertical and 5 horizontal moves.

3. **Expected Position of Each Ant:**
   - Each ant starts at a corner of the rectangle.
   - After 10 seconds, each ant will have moved 5 units vertically and 5 units horizontally on average.
   - This means the expected position of each ant will be 5 units away from its starting position in both the vertical and horizontal directions.

4. **New Dimensions of the Rectangle:**
   - The original dimensions of the rectangle are $20 \times 23$.
   - After 10 seconds, the expected new dimensions will be:
     \[
     (20 - 2 \times 5) \times (23 - 2 \times 5) = 10 \times 13
     \]
   - This is because each ant moves 5 units away from its starting position in both directions, effectively reducing the length and width by 10 units each.

5. **Expected Area of the Convex Quadrilateral:**
   - The expected area of the convex quadrilateral formed by the ants is the area of the new rectangle:
     \[
     10 \times 13 = 130
     \]

The final answer is $\boxed{130}$.