A space is said to be convex if the unit ball is a convex set.
This definition gives the intuition about the convex normed space. Now I am wondering what would be the similar definition for a strongly convex space?
P.S. This is more related to Randers-Finsler spaces.
I don't think the term "strongly convex" is commonly used for normed spaces. But if it was, then by analogy with strongly convex functions, which satisfy $$ f\left(\frac{x+y}2\right) \le \frac{f(x)+f(y)}{2} - c\|x-y\|^2 $$ they would be "spaces with quadratic modulus of convexity".
The modulus of convexity is defined as the function $${\displaystyle \delta (\varepsilon )=\inf \left\{1-\left\|{\frac {x+y}{2}}\right\|\,:\,\|x\|=\| y\| = 1,\ \|x-y\|\geq \varepsilon \right\},}$$ The space is uniformly convex if $\delta(\varepsilon)>0$ for every $\varepsilon \in (0, 2]$. This is already stronger than "strictly convex". A stronger property yet is having quadratic modulus of convexity, which means there exists $c>0$ such that $$ \delta(\varepsilon) \ge c\varepsilon^2, \quad \forall \ \varepsilon \in (0, 2] $$ This is the strongest kind of convexity that makes sense for a normed space. Examples of such spaces are $L^p$ and $\ell^p$ with $1<p\le 2$.