Convex Set- Intuition

Question

I acknowledge that "Convex Set" has no hollow between any two points in the set as shown in Wikipedia (https://en.wikipedia.org/wiki/Convex_set).

But I guess this intuition is based only on the three dimensional space. What if more than four? How the hollow is defined?

Thanks in advance!

Answer 1

Let $X$ be a vector space. Then a set $D \subseteq X$ is called convex if

$$ \forall x,y \in D, \lambda \in [0,1]: (1-\lambda) x + \lambda y \in D ~~.$$

Intuitively this means that for any two points in the set, the line joining them is fully contained in $D$ as well.

I think this intuition works reasonbly well for more exotic spaces than $\mathbb{R}^3$. In any case, working with the definition is what gives the right answers.

Answer 2

Just to give some graphical intuition, here's a plot of the line segment given by $\lambda x + (1-\lambda)y $ for two vectors $x$ and $y$:

from matplotlib import pyplot as plt
import numpy as np

fig = plt.figure(figsize=(10, 8))
ax = fig.add_subplot(111)
plt.xlim(-2, 10)
plt.ylim(-2, 10)

x = np.array([1, 7])
y = np.array([8, 5])

for λ in np.linspace(0, 1, 20):
  s = λ*x + (1-λ)*y
  ax.quiver(s[0], s[1], angles='xy', scale_units='xy', scale=1, width=0.003)