In this paper we consider the following analog of Bezout inequality for mixed volumes: V(P1,…,Pr,Δn−r)Vn(Δ)r−1≤i=1∏rV(Pi,Δn−1) for 2≤r≤n.
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We show that the above inequality is true when
Δ
is an
n
-dimensional simplex and
P1,…,Pr
are convex bodies in
Rn
. We conjecture that if the above inequality is true for all convex bodies
P1,…,Pr
, then
Δ
must be an
n
-dimensional simplex. We prove that if the above inequality is true for all convex bodies
P1,…,Pr
, then
Δ
must be indecomposable (i.e. cannot be written as the Minkowski sum of two convex bodies which are not homothetic to
Δ
), which confirms the conjecture when
Δ
is a simple polytope and in the 2-dimensional case. Finally, we connect the inequality to an inequality on the volume of orthogonal projections of convex bodies as well as prove an isomorphic version of the inequality.